"A=2πrh is the Surface Area for a Cylinder": Figurative and Operative Thought with Formulas

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“A=2πrh is the Surface Area for a Cylinder”: Figurative and Operative Thought with Formulas
Irma E. Stevens
University of Michigan
Stevens, I. E. (2022). “A=2πrh is the surface area for a cylinder”: Figurative and operative
thought with formulas. In Karunakaran, S. S., & Higgins, A. (Eds.) Proceedings of the
24th Annual Conference on Research in Undergraduate Mathematics Education. Boston,
MA. (pp. 613–621)
Available at: http://sigmaa.maa.org/rume/RUME24.pdf

“A=2πrh is the Surface Area for a Cylinder”: Figurative and Operative Thought with Formulas
Irma E. Stevens
University of Michigan
Researchers studying students’ quantitative reasoning argue that students’ reasoning
covariationally is essential for their construction of productive meanings for various ideas
related to relationships (e.g., linear, quadratic, trigonometric) and representations (e.g., graphs,
tables). In this report, I build on that literature by considering a unique feature of geometric
formulas, the idea that one formula can be used to re-present multiple (geometric) relationships.
To do so, I use The Formula Task in exploratory teaching sessions and a teaching experiment;
seven undergraduate students were given the formula “A=2πrh” and consider if/how the
formula can be used to re-present relationships in various dynamic geometric contexts. In
analyzing students’ responses, I use the ideas of figurative and operative thought to make sense
of the ways that students reason about and construct formulas. This report provides implications
for how students understanding formulas as ways to re-present situations vs. quantitative
relationships impact their interpretation of formulas.
Keywords: Cognitive Research, Teaching Experiment, Precalculus, Formulas
In the quantitative and covariational reasoning literature, there is evidence that supports that
students’ reasoning covariationally is essential for their construction of productive meanings for
rate of change (Carlson et al., 2002), trigonometric relationships (Moore, 2014), and numerous
other topics (e.g., Ellis et al., 2013; Johnson, 2015, 2013; Trigueros & Jacobs, 2008). Many of
these studies include students reasoning with dynamic contexts to construct various
representations. In this report, I build on that literature, specifically the ideas of figurative and
operative thought, by considering the idea that one formula can be used to re-present multiple
(geometric) relationships (I use re-present to emphasize an image being presented again a new
context (see von Glasersfeld, 1982, who describes re-present similarly as images that result
absent of perceptual material). To do so, I use The Formula Task, in which students are given the
formula “A=2πrh” and asked (i) to describe a situation that the formula can re-present, and (ii)
when given various dynamic geometric contexts, if/how the formula can be used to re-present
that relationship. This study is part of a larger study consisting of exploratory teaching sessions
and a semester-long teaching experiment (Steffe & Thompson, 2000) conducted with seven total
pre-service secondary mathematics teachers (heretofore students) aimed at understanding and
developing students’ meanings for formulas. To analyze the data from The Formula Task, I
consider the extent to which students’ activities relied on figurative or operative thought (Piaget,
1974). I report on the results of four students’ activities on the task. I then provide insights into
how these students’ reasonings show that students constructing or interpreting a formula as a re-
presentation of a relationship between quantities, rather than as an individual feature associated
with a particular shape, is a powerful way for students to reason.
Background Literature
Researchers have identified students’ difficulties with reasoning about formulas. For
example, students reverse symbols re-presenting variables’ measures (e.g., Clement et al., 1981,
1981) and treat symbols as static, or fixed given referents (e.g., Dubinsky, 1991; Gravemeijer et
al., 2000; Musgrave & Thompson, 2014). These difficulties are problematic for undergraduate
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students, particularly those in Calculus and Differential Equations sequences, in which students
need to reason symbolically and re-present and reason with changing quantities via formulas and
equations. One way that researchers have recently considered supporting students’ productive
meanings for formulas is to incorporate opportunities for covariational reasoning—reasoning
about two quantities change together (Carlson et al., 2002)—and dynamic situations. Along with
the examples of covariational reasoning research mentioned in the introduction, some researchers
have specifically considered dynamic geometric environments to support symbolization. For
example, Fonger et al. (2016) used the context of an area of a rectangle growing in proportion
with middle school students to explore and symbolize quadratic growth, and Panorkou (2020,
2021) has studied students’ meanings for volume and area using dynamic geometric
environments. In designing the tasks used in the exploratory teaching sessions and teaching
experiment of this study (including The Formula Task described in this report), the literature on
these dynamic geometric situations and covariational reasoning informed how we might support
students’ meaningful construction of formulas as re-presenting quantitative relationships.
One primary finding in the literature relevant to this report is the distinction between
figurative and operative thought (Piaget, 1974, 2001; Steffe, 1991). In essence, figurative
thought is associated with thought that foregrounds sensorimotor actions that are subordinate to
perceptual (figurative) properties, and operative thought is not constrained by sensorimotor
experience (see Moore (2016)). Building on the work of static and expert shaping thinking
(Moore & Thompson, 2015), Moore et al. (2019) took up this distinction to make sense of
students’ graphing actions. For instance, they described a student struggling to re-present a
dynamic relationship between two quantities that resulted in a trace that traveled from right to
left, because “it’s backwards” to the usual left to right graphing activities typically done in the
classroom. In this way, the students’ actions were constrained by the sensorimotor experience of
a graph’s trace rather than foregrounding the quantitative relationship re-presented by that trace.
In this report, I use the findings from The Formula Task to describe how the constructs of
figurative and operative thought can be used to construct viable models of students’ actions when
constructing formulas. Specifically, I considered how Thompson’s (1985) distinction between
figurative and operative thought could relate to geometric formulas. Thompson (1985, p. 195)
said, “Any set of schemata can be characterized as figurative or operative, depending upon
whether one is portraying it as background for its controlling schemata or as foreground for the
schemata that it controls.” In this study on formulas, the set of schemata does not rely on
perceptual actions taken on perceptual material that can be operated on, such as tracing right to
left as Moore et al. (2019) described, or repeating partitioning activities across contexts as Liang
& Moore (2020) described. Rather, the schemata relies on associations of a symbol (or collection
of symbols) in a formula. Thus, I sought to answer what meanings for formulas do students have
when they see “A=2πrh”? Those meanings could potentially be serving as controlling schemata
in the background if they are tied to attributes/shapes (e.g., (surface) area of cylinder, circle,
spherical cap), in which case the student would be engaging in figurative thought. Alternatively,
if those meanings include the formula as re-presenting quantitative relationships that could be re-
presented in a variety of contexts (e.g., a linear relationship between height and area that could
be re-presented in a rectangle, cylinder, spherical cap, etc. depending on what quantities the
symbols re-present in the situation), then the student would be engaging in operative thought.
The results in this paper detail particular illustrations of how students engage in figurative and
operative thought when reasoning with the formula A=2πrh.
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Methods
The larger study was split into two parts: a round of 3-5 individual exploratory teaching
sessions with four pre-service mathematics teachers (heretofore, students) and semester-long
individual teaching experiments with three students (Steffe & Thompson, 2000). All the students
were either in their first or second semester of secondary mathematics teacher program at a large
public university in the southeastern U.S. Each student had completed a Calculus sequence and
at least two other upper-level mathematics courses (e.g., linear algebra, differential equations)
with at least a C in the course. Moreover, each student was either enrolled in or had completed a
spring semester content course exploring secondary mathematics topics through a quantitative
and covariational reasoning lens inspired by the Pathways Curriculum (Carlson et al., 2015).
For the exploratory teaching sessions, the teacher-researcher (TR) individually interviewed
all four students who expressed interest in the study after contacting the entire class about the
study. During the interviews, the TR presented the students with tasks via a semi-structured
clinical interview style pre-interview and a series of 3-5 interviews (Clement, 2000). These
interviews lasted about two hours each, and this report focuses on Charlotte, Kimberley, and
Alexandria’s final task of the sessions, The Formula Task (details in the next section).
For the subsequent teaching experiment, three students were selected from a different
semester of the aforementioned secondary content course by identifying students that exhibited
different ways of reasoning on a modified version of the MMTSM assessment (Thompson, 2012)
and a clinical interview. The students in the teaching experiment individually met with the TR
and a witness-researcher (WR) approximately weekly for a semester. In this article, I report on
Lily’s activities on The Formula Task, the last task of the teaching experiment.
The analysis done for this report involved ongoing and retrospective analysis through which
the TR built these models of students’ mathematics (Steffe & Thompson, 2000). It was during
the retrospective analysis of the exploratory teaching sessions that the ideas of figurative and
operative thought became a viable way to distinguish between students’ reasoning about
formulas in The Formula Task. The TR then went through the data again to identify different
instances of figurative and operative thought, definitions that were carried through in both
ongoing and retrospective analysis for the teaching experiment. The resulting definitions are
presented in the results along with sample illustrations.
The Formula Task
In The Formula Task (Figure 1), students receive the following prompt: “Describe a situation
in which the formula A=2πrh describes a relationship between quantities. How does your
situation describe that relationship?”. After the students gave their initial responses, they were
presented with the dynamic situations presented in Figure 1 one by one: a cylinder (height
varying), a rectangle (width varying), a parallelogram (one of the side lengths varying), a cone
(height varying), and a spherical cap (with radius of sphere constant, spherical cap varying up
until a hemisphere). All these situations except for the cone situation entail relationships that can
be re-presented by the formula A=2πrh, with A re-presenting (surface) area, and h re-presenting a
certain length quantity (height, width, length, etc.) and r re-presenting a radius.
The purpose of this task is to analyze the images (quantitative, covariational, or otherwise)
students evoke from A=2πrh, and to determine the importance of the particular structure,
ordering, and use of the letters/numerals to their construction of situations. In this way, I consider
the role of students figurative and operative thought. For instance, a student with a figurative
meaning may conceive the formula as the normative formula for the surface area of an open
cylinder, and only an open cylinder; the student would not describe any other situation with that
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formula. Alternatively, a student with an operative meaning may have a meaning for a formula
as re-presenting a linear relationship that can be identified in various contexts. Specific results
are detailed in the next section.
Figure 1. The initial prompt in The Formula Task and images of the follow-up shapes that the students were
presented with in a dynamic geometric environment.
Results
The results section is organized into four sections. Recall that The Formula Task asked
students to answer both if A=2πrh was an appropriate relationship for a given and how (or why
not). In the results below, students’ figurative and operative thought is distinguished based on
whether the students were reasoning about the formula as a whole (first two sections) or a more
nuanced referencing of particular (groups of) symbols in the formula (last two sections).
Figurative Thought with the Formula as a Whole
In the first example of figurative thought, a student considers a formula as an attribute of one
and only one shape. For A=2πrh, a student may conceive the formula as the normative formula
for the surface area of an open cylinder, and only an open cylinder; any other context could not
be described using the given formula. For example, after concluding that the formula A=2πrh
was appropriate for the cylinder context (using the normative definitions for each of the
symbols), Charlotte said there was not a way for her to make sense of writing A=2πrh for the
(surface) area of a parallelogram or cone; her argument for the cone is in the following excerpt.
Charlotte: No, because if it were the area, then, it would-it would be, I would just be saying
like it was this shape [pointing to the cylinder shape], but that shape is this shape with
those cut out [forming a cone shape within the cylinder shape], so I can’t say that.”
As seen above, Charlotte’s justifications for the appropriateness of writing A=2πrh was
rooted in comparing the shapes to one another rather than focusing on a potential quantitative
meaning the formula could re-present across different contexts.
Alexandria had similar reasoning with the spherical cap. When the TR confirmed that, in
fact, A=2πrh was an appropriate formula given that A=surface area of the spherical cap,
h=height of the spherical cap, and r=the radius of the sphere from which the spherical cap is
formed, she was perturbed. She stated, “That really bothers me because that doesn’t make sense
to me... That’s the surface area for a cylinder, but that’s [the spherical cap] not a cylinder”).
Like Charlotte, Alexandria’s justification for a formula being appropriate for a context relied
more on the shape being considered than the relationship between quantities re-presented in the
formula. For both students, the formula was uniquely associated with the cylindrical shape.
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Operative Thought with the Formula as a Whole
To engage in operative thought with a formula as a whole, a student considers the formula as
re-presenting a relationship between quantities identified in a context. As an example, we return
to Alexandria and her work with the spherical cap. Although her initial reaction was that the
formula could not re-present the area of the spherical cap, when she engaged in covariational
reasoning, she made sense of the implications that formula meant in terms of the relationship
between the height and surface area of the spherical cap. Considering successive equal changes
in height for the spherical cap (see Figure 2), she stated “that each change in area is the same as
before.” Although she was uncomfortable with this conclusion because of she had difficulty
comparing changes in area in the spherical cap, she returned to the formula and said, “If r is
staying the same, then you have a linear relationship, which is constant, by definition, so it would
have to be.” Thus, rather than relying on associating a formula with a particular shape like she
did at first, Alexandria considered what quantities the formulas re-presented in each context and
considered the (linear) relationship that the formula was re-presenting in each case. Her going
through the activity of comparing the changes in area within and across the two contexts
indicated that her focus shifted from associating a formula with a shape (i.e., a cylinder), a
figurative association, to reasoning with a formula as re-presenting relationships between
quantities in the given context (i.e., a spherical cap), which is operative thought.
Figure 2. Alexandria’s equal changes in height marked on the sphere for The Formula Task.
Figurative Thought with Quantities Re-presented within the Formula
In this second example of figurative thought, a student considers a formula for a shape as
comprised of quantities/formulas that are identified in that shape. In this case of A=2πrh,
students may associate 2πr with circles, and thus any context that contains a circle (e.g., cylinder,
cone, sphere) may be associated with the formula in one way or another. For instance, Kimberley
considered the 2π in the formula A=2πrh and thought both the cylinder and cone were potential
candidates, but not the parallelogram or the rectangle. Her thoughts on the parallelogram context,
particularly her interpretation of the 2π in the formula, are in the following excerpt.
Kimberley: I don’t know why-you wouldn’t have pi for the area of a parallelogram. I guess
the r is whatever you want it to be, but you wouldn’t have a two pi.
TR: So why wouldn’t you have a two pi?
Kimberley: Because I don’t know what it would represent in a parallelogram. Like, in a
circle, it’s because you can like divide a circle into two pi radii, but you don’t have
anything even here that you could do that with.
In the excerpt above, Kimberley argues that 2π is associated with measurements of radii in a
circle, but because there is no circle present, there is nothing with which to associate the 2π
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(Note: It would be possible to re-present the height of parallelogram with 2π (i.e., 2πr)). On the
other hand, the cylinder and cone contexts do contain circles, and thus Kimberley used the
circles to justify her conclusions about why A=2πrh could re-present these contexts. The
following excerpt shows her reasoning. (It is important to note that Kimberley remained unsure
whether 2πr, πr2, 2πr2 was the formula for calculating the area of a circle, so she used them
interchangeably throughout the interview. At this point, she had settled on 2πr2.)
Kimberley: I mean, they’d obviously, either one of these [cylinder of cone] makes more sense
than like a parallelogram or a rectangle.
TR: Okay, and why does it make more sense for these?
Kimberley: Because you got a circle and you know that, like, to find the area of a circle,
you’re gonna have two pi included.
TR: Okay. Right, and because these [cylinder and cone drawings] both have circles, then this
formula [A=2πrh] kinda makes more sense for those?
Kimberley: Mhm.
Although figurative associations such as associating πr2 with the area of a circle can be
useful in constructing a quantitative structure in a context, it is important to note the effects of
maintaining figurative associations in constructing formulas. Namely, the student may attempt to
identify perceptual features in the context and then attempt to incorporate formulas associated
with those figurative elements in their construction of a formula, resulting in a non-quantitative
formula. For instance, after the previous dialogue, Kimberley identified both the cone and
cylinder as including circles and a varying height, thought that only one should be able to be
associated with the formula A=2πrh, and struggled to decide which one was appropriate. She
continued reasoning figuratively with formulas by attending to the triangle shape that she noticed
in the cone (but not the cylinder). At one point, Kimberley tried to write a formula for the cone
using the formula for the area of the circle combined with the area of a triangle: A=2πr2∙ ½bh,
crossing out the b for the base of the triangle, because the base is “just the circle part of it” which
the 2πr2 already accounted for, leaving her with A=πr2h as the formula for finding the surface
area of a cone (Figure 3). Note that all her reasoning relied on associating formulas with shapes.
Figure 3. Kimberley’s constructed formulas for a cone and cylinder.
Operative Thought with Quantities Re-presented within the Formula
To engage in operative thought with quantities re-presented within the formula, a student
considers the formula as consisting of quantitative operations between quantities identified in a
context. For the formula A=2πrh, these quantitative operations could include a quantitative
operation (i.e., multiplication) between a quantity re-presented by 2πr and another quantity re-
presented by h that would result in the measure of a quantity re-presented by A. This way of
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thinking is how Lily thought about the formula A=2πrh. More specifically, the formula pointed
to covariational relationships between quantities; she could anticipate the relationship between
the values of the amounts of change between quantities using her formula. For instance, when
explaining why she anticipated that the amounts of change in area would be equal for equal
changes in height, she stated, “[I]f you add like this much more to the height, you would multiply
the height, that equal change in height, times two pi r, and that’d be the equal change in area.”
She could assimilate this covariational relationship with several different geometric shapes. For
example, Figure 4 shows her comparing the cylinder and cone contexts by identifying and
comparing changes in surface area (A) for equal changes in height (h), concluding that between
the cylinder and cone, the cylinder was the “winner” (i.e., the context in which A=2πrh was
appropriate) because she identified equal changes in surface area for equal changes in height
(unlike with the cone). In this way, Lily engaged in operative thought across situations fluidly.
Figure 4. Lily’s work on the Cylinder and Cone in The Formula Task.
Discussion
This report includes different ways in which figurative and operative thought might occur,
depending on what symbols, groups of symbols, or formulas the students are considering. It also
showed that, similar to Moore & Thompson’s (2015) expert shape thinking, figurative thought
may not always result in problematic conclusions (in the case of Charlotte and the cone), but that
there is potential that figurative thought could result in ways of thinking about formulas that do
not rely on re-presenting quantitative relationships (in the case of Kimberley’s construction of a
formula for the surface area of a cone). Lastly, although both Lily and Alexandria’s examples of
operative thought included covariational reasoning, the definition of operative thought deals with
students’ capacity to reason quantitatively (not necessarily covariationally) about the
relationships rather than relying of associations of formulas with figurative materials (shapes).
Alexandria, nevertheless, showed how opportunities for covariational reasoning with perceptual
materials might support operative thought. These results are important for researchers to consider
when thinking about how undergraduate students’ meanings for formulas impact ideas such as
what it means to take the derivative or antiderivative of formulas.
Acknowledgments
This paper is based upon work supported by the NSF under Grant No. (DRL-1350342). Any
opinions, findings, and conclusions or recommendations expressed in this material are those of
the authors and do not necessarily reflect the views of the NSF. I would also like to thank Kevin
Moore, Leslie Steffe, Amy Ellis, and Edward Azoff for their helpful feedback on this study.
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