Developing Students’ Reasoning about the Derivative of Complex-Valued Functions with
the Aid of Geometer’s Sketchpad (GSP)
Jonathan Troup
University of Oklahoma
In this paper, I share results of a case study describing the development of two undergraduate
students’ geometric reasoning about the derivative of a complex-valued function with the aid of
Geometer’s Sketchpad (GSP). My participants initially had difficulty reasoning about the
derivative as a rotation and dilation. Without the aid of GSP, they could describe the rotation and
dilation aspect of the derivative for linear complex-valued functions, but were unable to generalize
this to non-linear complex-valued functions. Participants’ use of GSP, speech, and gesture assisted
with discovering function behavior, generalizing how the derivative describes the rotation and
dilation of an image with respect to its pre-image for non-linear complex-valued functions, and
recognizing that the derivative is a local property.
Keywords: Amplitwist, Complex-valued function, Derivative, Dynamic Geometric
Environments (DGEs), Gesture
Introduction
In the calculus reform era, a main goal was to develop students’ conceptual understanding of
calculus by integrating algebraic and geometric reasoning (Asiala, Cottrill, Dubinsky, &
Schwingendorf, 1997; Lauten, Graham, & Ferrini-Mundy, 1994; Meel, 1998). While similar
research has been conducted in other mathematical content domains, there is less research in
complex analysis, which allows potential for geometric reasoning. The purpose of this research
was thus to address two research questions: What is the nature of students’ reasoning about the
derivative of complex-valued functions, and what role does GSP play in developing students’
geometric reasoning about the derivative of complex-valued functions (i.e., the amplitwist)?
For my work, I adopted the National Council of Teachers of Mathematics’ (NCTM, 2009)
reasoning definition, which is “the process of drawing conclusions on the basis of evidence or
stated assumptions” (p. 4), and is categorized as algebraic or geometric. Algebraic reasoning
involves symbolic manipulation, and geometric reasoning involves spatial elements. I also refer
to inscriptions, which Roth and McGinn (1998) define as “signs that are materially embodied …
and because of their material embodiment, inscriptions (in contrast to mental representations) are
publicly and directly available, so that they are primarily social objects” (p. 37).
Literature Review
In this section, I summarize literature on students’ reasoning in the realm of complex analysis,
and the benefits of DGEs on students’ reasoning. As these domains draw on gesture as evidence
for reasoning, I conclude this section by synthesizing related gesture research.
The Teaching and Learning of Complex Numbers
There has been a recent increase in educational studies focusing on complex numbers
(Danenhower, 2006; Harel, 2013; Karakok, Soto-Johnson, & Anderson-Dyben, 2014;
Nemirovsky, Rasmussen, Sweeney, & Wawro, 2012; Panaoura, Elia, Gagatsis, & Giatilis; 2006;
Soto-Johnson, 2014; Soto-Johnson & Troup, 2014). The results of these studies show that
participants struggle to reason geometrically about complex numbers. In brief, Panaoura et al.
found that high school students had difficulty transitioning between algebraic and geometric
𝑎+𝑖𝑏
reasoning, while Danenhower showed undergraduate mathematics majors could not convert 𝑐+𝑖𝑑
to exponential or Cartesian form. Applying Sfard’s (1991) duality principle. Karakok et al. saw
that teachers displayed a process/object duality of the Cartesian form, but only an operational
level of the exponential form. Harel (2013) noted that teachers were unable to attach a geometric
meaning to the addition and multiplication complex numbers.
However, reasoning via embodied cognition may help students reason geometrically about
complex analysis. Nemirovsky et al. (2012) observed that through the usage of a “floor tile”
representation of the complex plane and stick-on dots and string to represent complex numbers,
pre-service teachers discovered multiplying by 𝑖 corresponds to a rigid 90° rotation. SotoJohnson and Troup (2014) also found that undergraduates integrated their algebraic and
geometric reasoning while drawing a representative diagram.
Dynamic Geometric Environments
In contrast with complex analysis research, research on educational technology is abundant.
Some suggest that technology can cause harm by promoting overgeneralizations (Clements &
Battista, 1992; Olive, 2000) or an over-reliance on technology (Salomon, 1990). Alternatively,
technology can help students and teachers refine mathematical ideas (Arcavi & Hadas, 2000;
Barrera-Mora & Reyes-Rodríguez, 2013; Heid & Blume, 2008; Hollebrands, 2007; Jones, 2000;
Olive, 2000; Tabaghi & Sinclair, 2013; Vitale, Swart, & Black, 2014) without relying on
mathematical authorities such as teachers or textbooks. DGEs such as Geogebra have been
shown to help develop concepts related to real-valued differentiation (Hohenwarter,
Hohenwarter, Kreis, and Lavicza, 2008; Ndlovu, Wessels, and De Villiers, 2010). Salomon
(1990) suggested that DGEs may help students by providing multiple representations of
mathematical objects. Hollebrands (2007) and Olive (2000) contend that GSP makes abstract
ideas appear more concrete, which could help students ground reasoning in the physical
environment. For example, Tabaghi and Sinclair (2013) found that while interacting with an
eigenvector sketch in Sketchpad, students reasoned via gesture.
Gesture
Many (Alibali & Nathan, 2012; Goldin-Meadow, 2003; Keene, Rasmussen, & Stephan, 2012;
Roth, 2001) believe “gestures can be used as a window into what students in a classroom are
thinking” (Keene et. al., p.367). Others suggest diagrams and gestures inform each other
(Chȃtelet, 2000; Chen and Herbst, 2013). For example, a vector could express the result of the
multiplication of two complex numbers and a flick of the wrist could convey the associated
rotation and dilation (Soto-Johnson & Troup, 2014). Some also state gesture and speech form a
single, integrated system to support both visual and verbal content (Alibali & Nathan, 2012;
Goldin-Meadow, 2003; Keene et al., 2012; Roth, 2001). For example, Goldin-Meadows found
gesture aids memory and the ability to describe it, while restricting gesture hampers this ability.
Some literature suggests gesture can transform over time from primarily representative to
primarily pointing as students become more familiar with mathematical procedures (Alibali &
DiRusso, 1999; Soto-Johnson & Troup, 2014; Garcia & Engelke, 2012; Marrongelle, 2007;
Soto-Johnson and Troup, 2014). Garcia and Engelke (2012) also noticed that their undergraduate
participants gestured more frequently when they were stuck on a task. Vitale et al.’s (2014)
participants initially gestured to remind themselves of a geometric concept, but transitioned into
using gesture as a validation tool. The students additionally interacted with virtual
representations of spatial gestures, which resembled their “real” gestures.
Given that gesture may result from explorations with technology (Tabaghi & Sinclair, 2013) and
that gesture can arise organically, it was critical to adopt a framework that would be sensitive to
this phenomenon. Thus, I employed embodied cognition as my theoretical perspective, which
allowed me to interpret my participants’ reasoning via gesture and actions taken within a DGE.
In the next section I summarize my interpretation of embodied cognition.
Theoretical Perspective
Embodied cognition is mainly concerned with the relationship between reasoning and actions
within the physical environment (Anderson, 2003), though the nature of this relationship differs
between researchers. While some suggest embodied cognition is a tool for mental cognition,
(Alibali & Nathan, 2012; Lakoff & Nuñez, 2000; Wilson, 2002), others dispense with mental
models entirely by equating reasoning with body-based activity, such as is found in
Nemirovsky’s (2012) fluid “realm of possibilities.” Under the second view, bodily experience is
itself an inextricable part of the learning process. So, one view is concerned with mental models
influenced by embodied action, while the other focuses on personal experience. These views do
not seem entirely incompatible; an impersonal description of pain as “the firing of C-fibers”
(Nemirovsky et al., 2012, p. 2) neither invalidates nor takes precedence over someone’s personal
experience with pain. Similarly, postulations about the cognitive way the mind responds to
bodily actions may not be contradictory to inferences made about a learner’s experience. I align
more closely with this second viewpoint. Similar to Soto-Johnson and Troup (2014c), I interpret
embodied cognition to include bodily actions taken within the physical environment. I believe
perceptuo-motor activity (including DGE activities) can influence reasoning and that reasoning
can influence bodily actions. Rather than postulate any cognitive mechanisms, I simply suggest a
relationship between how my participants utilized DGEs, gestures, and inscriptions. Particularly,
participants reasoned with GSP via construction and manipulation of geometric inscriptions and
via the production of virtual gesture through mouse movement. Thus, I take this reasoning and
participants’ usage of GSP, gestures, and inscriptions to be inseparable.
Methods
This report is a summary of my Ph.D. thesis. Thus, in this section, I describe some of the aspects
of research performed, including participant selection, task design, and data analysis.
Setting and Participants
To help with participant selection and triangulation, I attended complex analysis class sessions
related to the derivative. I video-recorded these classes and took notes paying particular attention
to gestures employed. Four students agreed to participate in my study, who I refer to as
Christine, Zane, Edward, and Melody. I conducted two sets of interviews, one with Christine and
Zane, and another with Melody and Edward. Each group worked on tasks over four nonconsecutive days. Each interview lasted two hours.
Concept Analysis of Amplitwist
Within the context of real-valued functions, the derivative function has a well-recognized
geometric interpretation as the slope of a tangent line. However, as the graph of a complexvalued function is four-dimensional, the generalization of this concept is not straightforward. To
help overcome this problem, one can represent the graph of a complex-valued function as two
sets of axes: one for domain and one for range. Thus, graphing a complex-valued function can be
represented as a transformation 𝑓: 𝐶 → 𝐶. In describing the derivative of a complex-valued
function geometrically, Needham (1997) considers how a complex-valued function maps an
extremely small circle centered about the point 𝑧 in the domain. “The length of 𝑓 ′ (𝑧) must be the
magnification factor, and the argument of 𝑓 ′ (𝑧) must be the angle of rotation,” (p. 196) a
concept that Needham refers to as an amplitwist. Needham further elaborates that the derivative
of a complex-valued function provides a linearization that locally approximates the function, as
“‘expand and rotate’ is precisely what multiplication by a complex number means” (p. 196).
The Tasks
The overall goal of the tasks I developed for my interview sequences was to encourage reasoning
about the derivative of a complex-valued function as described by Needham’s (1997) concept of
an amplitwist. I utilized tasks developed from a previously conducted pilot study.
In Tasks 1 and 2, students followed instructions on a lab worksheet to construct the function
𝑓(𝑧) = 𝑧 2 and 𝑓(𝑧) = 𝑒 𝑧 with the aid of GSP and predict how the function maps points, lines,
circles, and the complex plane (see Appendix B). The goals of this task were for participants to
establish proficiency with GSP and determine the mapping of circles under a complex-valued
function. For Task 3 I prepared the linear complex-valued function 𝑓(𝑧) = (3 + 2𝑖)𝑧 with a
complex-valued derivative in order to reduce the likelihood that students reasoned that the real
part of the derivative is a dilation factor and the imaginary part of the derivative is a rotation
factor. I asked participants to describe their geometric reasoning about the derivative of a
complex-valued function both with and without GSP, and to use 𝑓(𝑧) = (3 + 2𝑖)𝑧 to
demonstrate. I re-introduced GSP later to allow them to test their conjectures and continue to
explore their reasoning. The purpose of this task was to help participants relate the magnitude
and argument of the derivative to the way the linear function dilates and rotates a circle. In Task
4, I asked participants to generalize their reasoning from Task 3 to the functions from Tasks 1
1
and 2 as well as the functions 𝑓(𝑧) = 𝑧 2 , 𝑓(𝑧) = 𝑒 𝑧 , and 𝑓(𝑧) = 𝑧. The goal of this exercise was
to support students’ efforts to generalize the geometric reasoning they developed in Task 3 to the
general case. Finally, in Task 5, I asked students to determine the value of a derivative at a
(2𝑧+1)
particular point for the rational transformation 𝑓(𝑧) = (𝑧+𝑖)(1−𝑧). I gave them only geometric
information about a rational transformation, and asked them to use this information to
reconstruct the algebraic formula. The goal of this task was to encourage students to develop
reasoning about the derivative as a local property and to develop reasoning about points of nondifferentiability as they relate to the amplitwist concept.
Data Collection and Analysis
To obtain data, I video-recorded class sessions the instructor deemed relevant. I recorded all
interviews with both a camera to capture gesture and screen-capture software to record actions
taken with GSP. My analysis began by watching the videos in conjunction with the screencaptured GSP recordings to determine where the participants appeared to be making progress
toward a conception of the derivative of a complex-valued function as a local linearization. I
transcribed all recorded gesture, speech, and usage of inscriptions, after which I coded lines as
algebraic or geometric based on this data. Then, I wrote summaries of each day for each set of
participants wherein I detailed the progression of selected events.
Finally, I performed both a cross-case and within-case (Merriam, 2009; Patton, 2001) analysis on
the written summaries, referencing both the Excel spreadsheet and the actual raw data as needed.
Results and Discussion
My findings suggest that GSP helped my participants generalize how the derivative describes the
rotation and dilation of an image with respect to its pre-image for non-linear complex-valued
functions. Furthermore, asking participants to construct an algebraic formula from geometric
data served as a reminder that the derivative is a local property.
For the first two tasks, I did not ask the participants anything about the derivative. Rather, I
asked them to construct the function 𝑓(𝑧) = 𝑧 2 for Task 1, and 𝑓(𝑧) = 𝑒 𝑧 . In the process, I
answered their questions about how to use GSP, and asked them questions about how the
function mapped various circles. During this questioning, both groups noted that circles mapped
to other roughly circular shapes, and recalled that t points “rotate” and “dilate” or “stretch” when
multiplied by a complex-valued function. They did not appear to say anything about rotation and
dilation in conjunction with the derivative of a complex-valued function, however. Rather,
Christine states simply, “I don’t know like what slope means in complex world,”
When participants reasoned about the geometry of a linear complex-valued function 𝑓(𝑧) =
(3 + 2𝑖)𝑧 in Task 3, they verbalized that the function maps a circle to another circle which is
rotated by the argument of 3 + 2𝑖) and dilated by |3 + 2𝑖| with respect to the original circle.
While Melody and Edward made this observation in Task 3, Christine and Zane did not verbalize
this same point until Task 4, although they did mention that the “stretch” and “rotation” factors
do not change (see Error! Reference source not found.(a)). Melody and Edward observed, “the
derivative is rotating this consistently wherever it is and then it’s expanding it out whatever the
length of the derivative is. I guess we can see if that’s true” (see 1(b)).
(a)
(b)
Figure 1: 𝒇(𝒛) = (𝟑 + 𝟐𝒊)𝒛 transforms green input circle and spokes to blue output circle and
spokes of corresponding color (a) and Edward transforms a circle under 𝒛 → 𝟐𝒛 (b).
Later, during Task 4, Melody adds gesture to their description:
Melody: Like the center (points at (2,0)) would be at 2? So the center of the circle doesn't
necessarily depend on the derivative. Like where the output one is located doesn't depend on the
derivative (curls fingers slightly into claw (see Figure 2(a)), beats toward screen while moving
arm counterclockwise in an upper circular arc) The output one is just the size (hand makes claw
shape, extends fingers outward and back in) of it and (twists hand first clockwise like a
doorknob (see Figure 2(b)) then back counterclockwise) the rotation not the, like the location
would depend on the z squared.
(a)
(b)
Figure 2: Melody produces a “claw-like” gesture for dilation (a) and produces a “doorknob”
gesture for rotation (b).
One of the most interesting findings resulted from Task 5, wherein participants utilized
geometric data to construct an algebraic formula for a rational function. In this task, Edward and
Melody explicitly connected strange output behavior to non-differentiable points, stating, “that
can’t be differentiable there…It’s weird” (see Figure 3). At this point, Edward was able to
verbally reason geometrically about what made a point differentiable, by stating, “okay, so to
know if this is differentiable, we want to kind of know when, where, goes to, small circle goes to
small circle.” This utterance is an atypically precise geometric description of the fact that the
derivative is a local property. It was during this task that participants most precisely
characterized the derivative as a local property. Note that only Edward and Melody
accomplished this task, as Zane and Christine progressed more slowly through the tasks.
Figure 3: Edward and Melody zero in on a non-differentiable point.
Overall, I answer the two research questions listed above in the following ways. To answer the
first, I argue that my participants reasoned about the derivative of a complex-valued function via
embodied cognition in three distinct ways. In particular, they grounded their algebraic and
geometric inscriptions via gesture and speech as in Goldin-Meadow (2003) (see leftmost cycle in
Figure 4), integrated their algebraic and geometric reasoning methods via these inscriptions as in
Soto-Johnson and Troup (2014) (see center cycle in Figure 4), and grounded these reasoning
methods in both the real and virtual environments as in Hollebrands (2007), Olive (2000), and
Tabaghi and Sinclair (2013) (see rightmost cycle in 4. To answer the second question, I detail
three developments in reasoning which arose during their work with GSP, and seemed critical to
my participants’ reasoning about the derivative.
Figure 4: Integration through Embodied Cognition.
These three developments are as follows. First, my participants recognized a need for a
geometric characterization of linear complex-valued functions, which was supported by
grounding algebraic and geometric inscriptions via gesture and speech. Many of these
inscriptions were displayed by GSP. This development allowed participants to begin extending
their reasoning about real-valued functions to the complex-valued case. Second, my participants
reasoned geometrically that linear complex-valued functions rotate and dilate every circle by the
same amounts 𝐴𝑟𝑔(𝑓 ′ (𝑧)) and |𝑓 ′ (𝑧)|, respectively. Finally, the participants observed small
circles map to small circle-like objects under any complex-valued function.
Concluding Remarks
Thus, it appears that in my case study, gesture, DGEs and speech really did work together to aid
students in developing reasoning about the derivative of complex-valued functions. I conclude
by describing some possible teaching implications and directions for future research.
Teaching Implications
The findings described above suggest a few implications for teaching the derivative of a
complex-valued function. First, it demonstrates potential learning trajectory for students seeking
to develop their geometric reasoning about the derivative of a complex-valued function. In
particular, my students first developed reasoning about the geometry of lines in ℂ, then reasoned
geometrically about a constant derivative in terms of rotation and dilation, and finally reasoned
about the need to reason specifically about small circles. Second, it suggests that my students
worked beneficially with DGEs when placed in pairs and allowed some opportunity for free
exploration of related algebraic and geometric reasoning, as in Olive (2000). Finally, my
research suggests that it may be helpful to direct students’ focus to key points over the course of
their mathematical investigations, even when aided by a dynamic geometric environment (DGE).
Future Research
One possible direction for future research is to increase the breadth of these results by
implementing dynamic geometric environments (DGEs) on a large scale in real classrooms and
collecting quantitative data on student performance on tasks related to reasoning about the
derivative of a complex-valued function as an amplitwist. Such research would theoretically
allow these or related results to achieve some level of generality. Another possible direction is to
increase depth of the results in this case study by iterating on the last task specifically. My
participants reported beneficial effect from constructing algebraic information from exclusively
geometric information from GSP, so further research on the effects of similar tasks for other
students could prove highly interesting.
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