experts’ construction of mathematical meaning for

Experts’ Construction of Mathematical Meaningfor Derivatives and Integralsof Complex-Valued Functions

Michael Oehrtman1& Hortensia Soto-Johnson2

& Brent Hancock3

Published online: 9 July 2019# Springer Nature Switzerland AG 2019

AbstractWe engaged five mathematicians who conduct research in the domain ofcomplex analysis or use significant tools from complex analysis in their re-search in interviews about basic concepts of differentiation and integration ofcomplex functions. We placed a variety of constructivist, social-constructivist,and embodied theories in mathematics education in conversation with oneanother to explore the development of the expert participants’ construction ofmathematical meanings while moving between varying levels of abstractionfrom embodied concepts and real-world contexts to symbolic manipulationand formal theories. The mathematicians relied heavily on direct applicationof concepts and analogies from differentiation of real-valued functions andemployed rotation and dilation as a local linear description of the action of acomplex differentiable function with attendant repeated mental imagery andphysical gestures. They also employed reasoning about real-valued line integralsto interpret contour integrals but acknowledged significant struggle to concep-tually interpret what was analogously accumulated in the complex case. Instead,they all developed more personal meanings through a process of reconcilingvarious aspects across their concrete to formal domains of reasoning. Much ofthe observed construction of meaning was manifested through contextualizingwell-understood aspects of formal mathematical theory. We consequently ex-plore implications for characterizing mathematical conceptual development as aninterplay between concrete and formal reasoning rather than a developmentfrom one to another.

Keywords Abstraction . Complex analysis . Contextualization . Expert reasoning .

Derivative . Integral

International Journal of Research in Undergraduate Mathematics Education (2019) 5:394–423https://doi.org/10.1007/s40753-019-00092-7

* Michael [email protected]

Extended author information available on the last page of the article

http://crossmark.crossref.org/dialog/?doi=10.1007/s40753-019-00092-7&domain=pdf

http://orcid.org/0000-0003-4101-5762

mailto:[email protected]

Introduction

As a result of the calculus reform movement many calculus text authors wrote oredited their texts to promote geometric interpretations of analytic concepts. Au-thors of textbooks in other content areas, such as linear algebra, differentialequations, abstract algebra, and statistics followed suit. In contrast, little workhas been done to emphasize the rich geometric structure in introductory complexanalysis courses. For example, authors of complex analysis texts generally intro-duce the definition of the derivative of a complex-valued function f at the point z0

as the complex limit f 0 z0ð Þ ¼ limz→z0

f zð Þ− f z0ð Þz−z0

if it exists, without any geometric

connections. Although this algebraic inscription is introduced as early as chaptertwo of many texts (Brown and Churchill 2009; Paliouras and Meadows 1990; Saffand Snider 2003), authors claim “geometric interpretations of derivatives offunctions of a complex variable are not as immediate as they are for derivativesof functions of a real variable” (Brown and Churchill, p. 59) or “… there is littlesimilarity between the interpretations of quantities such as f 0 xð Þ and f 0(z)” (Zilland Shanahan 2013, p. 121). Geometric interpretations tend to be delayed untilafter conformal mappings are presented near the end of the text, and thus are notleveraged as a foundational interpretation and potentially not even addressed inintroductory courses. Similarly, integration of complex-valued functions is intro-duced using algebraic inscriptions similar to those employed in a multi-variablecalculus course. If w : [a, b]→C is a complex-valued function of a real variablesuch that w(t) = u(t) + iv(t), where u and v are real-valued, then

∫ba w tð Þdt ¼ ∫ba u tð Þdt þ i ∫ba v tð Þdt, provided the integrals exist. A contour integralover a path γ : [a, b]→C of a function f :C→C is further defined as

∫γ f zð Þdz ¼ ∫ba f γ tð Þð Þγ 0 tð Þdt. If geometric representations are provided, they areoften for the integrals of u and v in Euclidean space rather than leveraginggeometry to visualize w or f and γ.

Given complex analysis textbooks rarely develop foundational geometric interpre-tations of either differentiation or integration on the complex plane, students enrolled incomplex analysis courses may not have opportunities to develop rich visualization ofeither of these concepts. In contrast, mathematics policy documents have long stressedthe importance of blending algebraic and geometric reasoning in K-16 mathematics(Council of Chief State School Officers & National Governors Association Center forBest Practices (2010)). For example, the Mathematical Association of America’sCUPM Guidelines recommends that “mathematical science major programs shouldpresent key ideas from complementary points of view: continuous and discrete;algebraic and geometric; deterministic and stochastic; exact and approximate”(p. 12). They also stress that “geometry and visualization are different ways of thinkingand provide an equally important perspective … [which] complement[s] algebraicthinking … [and] remain[s] important in more advanced courses” (p.12). As such, inthis exploratory study, we felt it worthy to investigate the research questions:

1. What geometric interpretations do mathematicians use to reason about differenti-ation and integration of complex-valued functions?

International Journal of Research in Undergraduate Mathematics Education (2019) 5:394–423 395

2. How do embodied, symbolic, and formal aspects of mathematicians’ reasoninginteract?

Literature Review

In this section, we provide a brief historical account of the development of complexalgebra and analysis, which parallels the emphasis on symbolic reasoning provided inmost complex analysis texts. We also summarize the current literature on the teachingand learning of complex analysis.

Historical Development of Complex Numbers and Functions

Given the lag in development of geometric interpretations from algebraic uses of

the inscription i ¼ ffiffiffiffiffiffi−1

p, it may not be surprising that authors of complex analysis

texts emphasize symbolic reasoning about analytic concepts. The symbolffiffiffiffiffiffi−1

pwas introduced for mathematical problem-solving competitions in the early six-teenth century, when even the validity of negative numbers was questioned. Nahin(1998) recounts Scipione del Ferro’s secret technique for solving the depressed

cubic, x3 = px + q, which involved manipulating expressions withffiffiffiffiffiffi−1

pand

reducing to real-valued solutions. Although able to perform arithmetic operationseffortlessly with these complex roots, del Ferro remained perplexed by their veryemergence and by their meaning. Gerolamo Cardano and his student LodovicoFerrari extended and published these techniques, but remained skeptical of theirgeneral validity due to the lack of a physical interpretation. Over a century later,

John Wallis offered a first interpretation offfiffiffiffiffiffi−1

pas the mean proportional of two

line segments in a geometric construction. Not until an additional century hadpassed did Caspar Wessel first characterize the complex number a + ib as a point(a, b) in the complex plane or the “directed radius vector from the origin to thatpoint” (Nahin 1998, p. 49). Reasoning about complex numbers as vectors allowedWessel to further imbue a geometric interpretation to their arithmetic, much asthey are understood today. For example, the product of two complex numbers zand w results in a rotation and dilation of the vector w by the argument andmagnitude of the complex number z, respectively.

In the early 1800’s, Cauchy introduced complex function theory, beginning with anextension of the derivative idea to complex variables. This work led to the Cauchy-Riemann equations and Cauchy’s First Integral Theorem, derived algebraically withoutgeometric motivation, as depicted in many complex analysis texts. Gauss’s work,including identifying conditions for conformal maps, popularized geometric interpre-tations of the complex plane and adoption of the now-standard a + ib notation.Riemann’s dissertation in 1851 expanded these geometric interpretations significantly,developing branched coverings of domains in the complex plane, the relationship of thederivative to the geometry of conformal maps, and techniques employing geometricallyinspired choices of coordinate systems. Riemann’s work and geometric approach led tocontinued work on his Riemann mapping theorem as well as the study of algebraic andcomplex manifolds.

396 International Journal of Research in Undergraduate Mathematics Education (2019) 5:394–423

Much of the geometric insight behind these early and modern advances are rooted inthe interpretation of the derivative as a local linearization f (z) ≈ f (z0) + f 0(z0)(z − z0), alinear mapping in terms of multiplication by a complex number f 0(z0), and multiplica-tion by f 0(z0) = reiθ as a rotation by angle θ and a dilation by factor r in the complexplane. Such interpretations are often delayed in introductory complex analysistextbooks until late chapters on conformal maps and Riemann surfaces. Needham’s(1997) Visual Complex Analysis is a notable exception, developing a foundationalgeometric interpretation of the derivative, and subsequently integrals, of complexfunctions. Needham introduces complex differentiation by revisiting the derivative ofa real-valued function f :R→R through a lens of a linear mapping on infinitesimalvectors dx at x0, expanding or contracting by the scalar factor f

0(x0). Extending this ideato the geometric interpretation of multiplication by a complex number, Needhaminvokes the notion of an “amplitwist,” writing the “local effect” of an analytic function“on an infinitesimal disc centered at z is, after translation to f (z), simply to amplify andtwist it. The ‘amplification’ is the expansion factor, and the ‘twist’ is the angle ofrotation…

f 0 zð Þ ¼ the amplitwist of f at z¼ amplificationð Þ ei twistð Þ

¼ j f 0 zð Þjei arg f 0 zð Þ½ �” p:195ð Þ:

In order to provide a geometric interpretation of the complex line integral ∫K f zð Þ dz,Needham (1997) breaks the curve K in the domain into segments Δi with midpoints ziand defines the integral as the limit of Riemann sums ∫K f zð Þdz ¼ limn→∞∑n

i¼1 f zið ÞΔi.Given this definition centrally involves multiplication, each Δi may be viewed asrotated and dilated by the argument and magnitude of f (zi). Finally, these image vectorsare summed in the co-domain to obtain the resultant vector. In the next section, wesummarize research that suggests students can adopt geometric reasoning about thearithmetic of complex numbers to the derivative of complex functions, as well as otherrelated literature.

Research on Teaching and Learning Complex Analysis

Much of the literature on teaching and learning of complex analysis is limited toexploring undergraduates’ and inservice teachers’ geometric reasoning of arithmeticand algebraic concepts of complex numbers (Danenhower 2000, 2006; Harel 2013;Karakok et al. 2015; Nemirovsky et al. 2012; Panaoura et al. 2006; Soto-Johnson2014; Soto-Johnson and Troup 2014). This work illustrates undergraduate students’and inservice teachers’ tendency to prefer algebraic reasoning even when geometricreasoning simplifies the task. On the other hand, Nemirovsky et al. demonstratedhow undergraduates’ geometric reasoning of the arithmetic of complex numberscould be developed by engaging in physical activities. Soto-Johnson (2014) andSoto-Johnson and Troup (2014) found similar results with high school students andundergraduates who used a Dynamic Geometric Environment (DGE) to explore thegeometric meaning of the arithmetic of complex numbers in terms of transformationson the complex plane.


Some researchers have begun to explore analytical concepts related to complexvariables (Soto-Johnson et al. 2016; Troup et al. 2017) but the work in this area remainsscant. Troup et al. (2017) showed how DGEs can assist with developing undergradu-ates’ geometric reasoning about the derivative of complex functions. With the aid oftechnology, participants abandoned their misconception that the derivative of a com-plex function represents the slope of a tangent line. More importantly, they came torecognize the derivative of a complex function as a dynamic amplitwist. Soto-Johnsonet al. (2016) documented mathematicians’ conceptions of continuity of complex-valuedfunctions through metaphors stemming from prior physical embodied experiences.Although the formal definition of continuity requires codomain-first reasoning, severalof the experts’ metaphors were predominantly domain-first interpretations.

While there is abundant literature on students’ understanding of derivatives anddefinite integrals in introductory calculus of real variables, little of it touches oninterpretations that are productive foundations for geometric interpretations of thecorresponding concepts in complex analysis. Neither standard curricula nor educationalresearch tend to address the derivative of a function from R→R as a linear map as apotential precursor to the amplitwist concept. Introductory multi-variable calculus alsodoes not typically provide such a foundation, and instead frames the derivativeprimarily in terms of partial derivatives or equations of tangent planes. Even more,most texts (quite reasonably) rely on the independence of component functions for real-variable derivatives to simplify the situation to multidimensionality only in the domain.Although real-valued line integrals provide a possible foundation for the complex case,little education research has focused on the related concepts. Multiplication in theintegrand of a contour integral by a complex-valued γ 0(t), however, introduces arotation of the real and complex components of f that does not have a direct real-valued analog.

Theoretical Perspective

Our theoretical framing for this research leverages the fundamental mechanism ofconceptual innovation identified by John Dewey. Dewey viewed novel meaning asan emergent characteristic of inquiry from the dialectic interactions among intentionalselection, application, and evaluation of conceptual tools against problematic situations(Hickman 1990). Not only might such investigation produce new insight regarding theproblem, but the tools themselves owe their shifting meaning to the anticipated andunanticipated effects they have in the inquiry. We first use this focus to place somecommon theories in mathematics education in conversation to draw upon relevantfeatures to characterize development of new mathematical meaning. In our analysis, weexplore the interplay between our experts’ reasoning, ranging from real world experi-ence to formal mathematical theory, through the lens of these constructs. Finally, in ourdiscussion, we reflect on implications of these observations for the various theories thatinspired our interpretations.

Dewey distinguished reflective inquiry from the bulk of non-reflective or routinehuman experience in which nothing has become problematic and one may operate oncommonplace understandings and expectations (Hickman 1990). When an aspect of asituation is taken as problematic, however, one’s engagement may become inquirential.


Inquiry is active in the sense that one purposefully selects potentially appropriate tools(here we focus on cognitive tools) to apply to the problem. These tools are then used toprobe the situation, thus changing it. Such inquiry becomes reflexively testable in thesense that the system provides feedback about the tool, idea, or language initiatingevaluation of the tool itself and its interaction in the system. The tool is thus changed asits meaning incorporates its perceived effects. Inquiry becomes productive in thisdialectic between the tool and problem, through which new ways of perceiving andunderstanding emerge. We now discuss various learning theories used in mathematicseducation with particular attention to these generative mechanisms.

Many mathematics education theories draw on neo-Piagetian interpretations ofabstraction. According to Piaget, as an individual progresses through sensorimotor,preoperational, concrete operational, and formal operational stages, they shift fromphysical actions on tangible objects towards developing abstract mental operations andsymbolism. His characterization of reflective abstraction consisted of both the devel-opment of goal-oriented schemes abstracted from an individual’s actions and coordi-nations of actions on already constructed structures as well as further cognitivereorganization of what has been transferred to function on new inputs and goals (VonGlasersfeld 1995, pp. 104–105). Dewey’s view of inquiry is reflected in Piaget’s focusof a learner engaged in goal-oriented activity and their anticipations of the effects ofemploying their schema. Assimilation of results consistent with one’s expectationsbrings new and more generalized meanings into a scheme, while the structure ofunanticipated results must be accommodated by reorganizing schema to incorporatenovel aspects of that structure. Furthermore, conscious awareness and analysis of one’sschema and construction process may result in new schema, that is “retroactivethematization,” of one’s own thought.

These reflective abstractions serve as the fundamental constructive mechanismin the Action Process Object Schema (APOS) framework (Asiala et al. 1996;Breidenbach et al. 1992; Dubinsky 1991). Within APOS, actions on existingmathematical objects are reflectively abstracted (interiorized) to become mentalprocesses, which are in turn reflectively abstracted (encapsulated) to form objects.Multiple objects can then be collectively organized to form a coherent cognitivestructure called a schema. Sfard (1992) formulates a similar progression fromoperational to structural conceptions but notes an inherent interdependence be-tween their development. Specifically, reification of a concept into an object relieson the internalization of operations upon it, while reciprocally, the ability toperform such operations requires an object on which to operate. Thus, whilereflective abstraction produces successively more abstract mathematical concepts,that process also restructures previous conceptual levels.

Tall (2013) employs similar developmental mechanisms but primarily decomposesmathematical thought in terms of the form of the thought rather than its developmentaltrajectory. He characterizes learning as a long-term endeavor that entails interactivelydeveloping “three worlds of mathematics.” The world of conceptual-embodimentresults from our perceptions and actions in the real world, yielding rich mental imageryrefined via verbal communication. The world of operational-symbolism stems fromembodied human actions which transform into symbolic procedures, and the axiomat-ic-formal world builds on formal axiomatic systems whose properties are establishedthrough mathematical proof. Although Tall stresses entwinement of conceptual


development between these three worlds, there is additionally a general progressionfrom embodied to symbolic to formal reasoning.

The 1970s rediscovery of Lev Vygotsky’s work in the west led to an infusion ofstudy of social influences in cognitive developmental processes (Vygotsky 1987;Wertsch and Tulviste 1992). Vygotsky adapted Piaget’s notion of “spontaneous repre-sentations” to refer to biologically rooted operations that develop through the opera-tions of a child’s own thought (p. 173). As such, “The strength of the everyday conceptlies in spontaneous, situationally meaningful, concrete applications, that is, in thesphere of experience and the empirical” (p. 220). Vygotsky posited that spontaneousconcepts follow a bottom-up development that manifests itself through a child’smediated transition from tools to signs. While tools are grounded in the physicalmanipulation of an external object, signs encapsulate abstract information and proper-ties of language absent from the original tool (Vygotsky 1987). On the other hand, thescientific concept develops in a more “top-down” manner; it originates “in the domainof conscious awareness and volition. It grows downward into the domain of theconcrete, into the domain of personal experience” (Vygotsky 1987, p. 220). Preciselydue to their opposing directions of development, Vygotsky argues that spontaneous andscientific concepts form reciprocal influences within the zone of proximal developmentto facilitate the social construction of knowledge.

The theory of Realistic Mathematics Education is often operationalized in termsof three design heuristics (Gravemeijer et al. 2000). The principle of reinventionuses students’ informal reasoning as a starting point and “suggests ways in whichstudents’ informal interpretations and solutions might ‘anticipate’ more formalmathematical practices” (p. 239). Emergent models characterize students’ engage-ment in activity and representation that progresses from concrete to abstract.Students’ “models are initially tied to activity in specific settings and involvesituation-specific imagery” (p. 243), but evolve along with their purpose to inter-pretations independent of situation-specific imagery, and ultimately conventionalmathematics independent of support from prior models and applicable “as an entityin its own right.” Though Gravemeijer, et al. acknowledge their levels “of activityclearly involve a developmental progression,” they note that activity may often“fold back” to prior levels. A key mechanism in this folding back may be the thirdheuristic of didactical phenomenology which focuses “on how mathematical inter-pretations make phenomena accessible for reasoning and calculation” (p. 240).Thus, not only are mathematical meanings abstractions of students’ more concreteactivity, but they are tools through which students may conceptually operate on andreconstruct that activity.

Although Piaget’s and Vygotsky’s work attend to the sensorimotor foundations oflearning, Piaget viewed development in terms of more advanced characteristics ofthought continually superseding and displacing less powerful forms. AlthoughVygotsky critiqued such a displacement view, favoring his dialectic characterizationof spontaneous and scientific thought, he did not fully develop the underlyingmechanisms due to his early death. Work in cognitive linguistics emphasized the roleof embodied cognition throughout all aspects of development, as exemplified in Lakoffand Núñez (2000) analysis of mathematical thought in Where Mathematics ComesFrom. Generally, embodied cognition is characterized as reasoning derived fromexperiences with our physical and social environment. For some, such as Lakoff and


Nuñez, this means that reasoning is based in perception and action, and their workfocuses on mental models described via metaphors. Others, such as Nemirovky et al.(2012), focus on actual physical processing as discussed in the literature review,arguing that learning is doing or moving in ways consistent with targeted mathemati-zation. Other researchers situate their embodied work in virtual environments. Forexample, researchers (Tabaghi and Sinclair 2013; Troup 2015; Troup et al. 2017) foundthat interacting with DGEs facilitates gesturing consistent with what one sees on thescreen or aligned with one’s mouse movements.

Working within the philosophy of language, Black (1962) argued that metaphorsand scientific models operate in an “interactionist” manner to produce new meaningssimilar to Dewey’s dialectic view of inquiry. Models and metaphors require one toconsider the implications of the comparison for the target domain and to reinterpret thesource domain in the process of applying it to the target. In a metaphorical example,one may conceive of a line segment connecting the origins of two parallel rays as ametaphorical triangle, but in doing so one cannot simply apply an antecedently formedconcept of triangle as-is. The subsequent mutual implications may foster ontologicalcreativity, leading, for example, to a rudimentary projective geometry. Invoking theexample of James Clerk Maxwell relying on intuition about incompressible fluids toreason about his new theories of electrical fields, Black argued that, “the maker of ascientific model must have prior control of a well-knit scientific theory if he is to domore than hang an attractive picture on an algebraic formula” (p. 239). Something newand actively responsive to the situation is required of all concepts involved.

Throughout our design, analysis, and interpretation, we drew freely from each ofthese theories of mathematical development and abstraction. Our primary focus wasthen drawn to interactions between embodied, symbolic, and formal mathematicalconcepts and how our expert participants drew conclusions about any of these domainsbased on reasoning within another. We particularly focused on reciprocal influencesbetween interacting domains to document such dialectic influences as particularly richsites for creation of new meaning.

Methods

Participants and Setting

This research is part of a larger study in which we explored mathematicians’ geometricreasoning about the arithmetic of complex numbers and analytic concepts of complex-valued functions. These mathematicians are the same participants from the work ofSoto-Johnson et al. (2016), thus we use the same pseudonyms. The five Ph.D.mathematicians, from three different institutions within the same state, participated ina 90-min, video-taped, semi-structured interview. Becky, Judy and Rafael’s area ofexpertise is complex analysis, Luke’s area of expertise is differential geometry, andAndrew’s background is in differential equations. All five of the mathematicians havetaught introductory complex analysis on a regular basis and four of the mathematicianshave published in this area. None of the participants used Needham’s text for theircourse; they adopted open-source textbooks or textbooks listed in the Introductionsection, which do not contain a geometric interpretation of complex differentiation and


integration. Although all but one participant was familiar with Needham’s termamplitwist, a majority of the participants had to reconstruct the concept, which weillustrate in the Results section. In this paper, we focus on the participants’ responses tothe questions about their geometric interpretation of complex differentiation andintegration. The questions were:

1. How might one geometrically explain the relationship of the Cauchy-Riemannequations to complex differentiability?

2. What does it mean geometrically that the derivative of f (z) = z2 is 2z?3. How might one explain integration of complex-valued functions geometrically?

As part of the interview, the participants had access to a black or white board and wereencouraged to use it as they saw fit. We probed when a response seemed unclear orwhen the description of their geometric reasoning was incomplete. Some participantsstruggled to answer the first question, thus we asked them to explain how they reasongeometrically about complex differentiation. This question prompted some participantsto employ the function f (z) = z2 on their own accord. In these cases, we informed theparticipants that we intended to ask them about the derivative of this function anddirected them to summarize their response.

Analysis

We began our analysis by transcribing all the interviews and documenting allgeometric language, geometric inscriptions, and gestures that were used by theparticipants to convey their response. In addition, we documented their contextualapplications and interpretations of complex functions, derivatives, and integrals. Wedocumented corresponding participants’ gestures, which Soto-Johnson and Troup(2014) showed are often integrated while reasoning about both geometric andalgebraic inscriptions. Given this finding and the fact that several participantsreasoned symbolically before reasoning geometrically, in the second round ofanalysis we documented the participants’ symbolic reasoning. Similar to the find-ings of Soto-Johnson et al. (2016), at times the participants introduced a metaphorto further elaborate on their response. As such, we found that in an attempt toanswer the prompts, our research participants integrated embodied, symbolic, andformal reasoning. This integration was not always smooth or complete, as weillustrate in the Results section. In the final stage of analysis, we created storylinesdocumenting the evolution of each participant’s response and identified cross-cutting themes. In the Results section, we place descriptions of gestures in paren-theses and verbiage articulated simultaneously with the gesture in bold.

We focused our analysis on segments of transcripts where the experts wereengaged in some aspect of problem-solving to identify inquirential characteristicsof problems and solutions, as defined by Dewey, and the emergence of meaningsassociated with that inquiry. In particular, we noted whether the experts framed theirvarious embodied, symbolic, or formal expressions of mathematical ideas as prob-lems or employed them as tools to resolve another problem. In these cases, we alsodocumented the reciprocal influences manifest from the experts’ expressed impli-cations or changes in the situation derived from applying tools as well as from their


evaluation of potential tools (either for viability prior to application or for effectafterward). We used these analyses to compose narratives of each individual’smeaning-making throughout the interviews then compared the five narratives forthemes.

Results

Differentiation

We categorized segments of the mathematicians’ responses to the differentiation ques-tions into comparisons to differentiation of real-valued functions, amplitwist models,and a variety of idiosyncratic images. While interpreting the complex derivative, all ofthe mathematicians relied heavily on comparisons to differentiation of functions fromR→R, R2→R, and R2→R2. Participants emphasized aspects of dependence on thedomain point, local behavior of the function, linear approximation of this behavior,errors approaching zero “more quickly” than ∣z − z0∣, multiplication by f 0(z0), anddecomposing the action into a rotation and a dilation. Four of the five mathematiciansultimately relied heavily on geometric interpretations consistent with Needham’s (1997)description of an amplitwist, drawing several pictures and gesturing the action. All of themathematicians represented these actions algebraically and graphically in Cartesiancoordinates, but several commented that polar representations are more appropriatefor rotation and dilation. They all drew coordinate grids in a domain plane and theirimages in a codomain plane to illustrate the local stretch and rotation in the mapping.Repeatedly, the mathematicians either moved their hands from one plane to the other onthe board or an imagined version in space. This gesture generally entailed spreadingtheir fingers apart to indicate stretching and twisting their wrist to indicate rotation, oftendoing so seemingly subconsciously as a natural embodiment of their symbolic expres-sions for complex multiplication.

All of the participants initiated conversations about how they explain concepts ofcomplex differentiation and integration to their students, initially emphasizing symbolicreasoning. After expressing the complex function in coordinates f (x + iy) = u(x, y) + iv(x, y)in order to examine the limit definition of the derivative, they considered paths alongcoordinate axes, partial derivatives, level curves, and matrix representations of linear maps.Below we summarize each of the participant’s responses to illustrate these findings, butcombine Andrew’s and Luke’s responses due to their similarity.

Rafael: Rafael began his discussion of complex differentiability and the Cauchy-Riemann equations by noting the technicalities of real differentiability: that differen-tiability may occur with non-intuitive geometry, the difference between differentiabilityat a point and on a neighborhood, and that the limit definition is “not very convenient.”He thus introduced “Lemma 1:… f is complex differentiable at a point if and only if f isreal differentiable at the point and the Cauchy-Riemann equations hold there,” and“Lemma 2:… f is real differentiable if the partial derivatives exist and are continuous atthe point of interest to us.” He noted that although Lemma 2 is technically powerful, hefelt “there is no real geometry” in it and that Lemma 1 provided more meaningfulinterpretations.


Developing from Lemma 1, Rafael invoked the real derivative as a linear approx-imation to the function which he expressed as Δw =Δu + iΔv, where

Δu ¼ a Δxð Þ þ b Δyð Þ þ ojzjΔv ¼ c Δxð Þ þ d Δyð Þ þ ojzj:

This led him to introduce the Jacobianmatrix as he mentioned that bothΔu andΔv can

“be described by multiplication of a complex number” if and only if the matrixa bc d

� �

has “certain symmetries” (rotates hand over the linear equations). This statement andgesture simultaneously invoke a linear mapping of a function f : R2 → R2 asmultiplication by a real 2 × 2 matrix, complex linearity of f : C→C as multiplicationby a complex number, and the geometry of rotation and dilation of multiplication bya complex number. In working out the details of this constraint, Rafael wrote thecomplex value of the derivative as L = A + iB, and complex linearity asmultiplication LΔz = (A + iB)(Δx + iΔy) to obtain the result Δu = AΔx − BΔy andΔv = BΔx + AΔy. He clarified that, “In an actual class, we would have actuallylooked at the matrix description of matrix multiplication at least once,” as he wavedhis hand over his original system of equations for Δu and Δv in terms of the matrixcoefficients a, b, c, and d. Rafael then returned to the new system for Δu and Δvobtained by expanding LΔz and concluded that the resulting “symmetries” yieldA = a = d and B = c = − b. Rafael explained that if f is real differentiable then thecoefficients are the partial derivatives of u and v with respect to x and y, resulting inthe Cauchy-Riemann equations.1

Rafael explained that he viewed the function as locally linear, Δw = LΔz + o|z|,while twisting his hand back and forth over the L in the equation (Fig. 1). He thenemphasized the physicality of this conception, saying “the whole disc (points toz-plane) is going to get multiplied by a complex constant [which] is going to rotatethe disc and expand it out.” As he articulated these ideas, he mimed grabbing a disc inthe z-plane, rotated and expanded it with his hands, and placed it into the w-plane asshown in Fig. 2. He went on to say that “it’s like having a turntable that you can spinand expand,” gesturing as though he was holding onto a steering wheel while turningand spreading his hands out to indicate the expansion.

When asked about the derivative of the function f (z) = z2, Rafael instantiated each ofthese images and physical actions to interpret the local complex linearity of f as anamplitwist. He began by reiterating that “differentiability of any mapping means that asmall patch of it can be approximated by what geometrically corresponds to a rotationand an expansion” (rotates and expands hand). Then he transitioned to symbolicreasoning as he wrote w = f (z) = z2 andΔw ≈ f 0(z0)Δz to facilitate discussing mappingsof “little patches” near z0 in the input plane. As a special case, Rafael considered the

1 Rafael chose to articulate these results mostly in the language of his systems of equations forΔu andΔv. In

the corresponding “matrix description”ΔuΔv

� �¼ a b

c d

� �ΔxΔy

� �, note that once we restrict a = d and c = − b,

this matrix multiplication geometrically yields the familiar rotation and dilation of the vectorΔxΔy

� �. Thus, this

rotation-dilation produced by matrix multiplication has the same geometric effect as multiplying the complexnumber Δz =Δx + iΔy by the complex number L = A + iB.


point z0 on the unit circle and on the real axis, which he remarked would be scaled by afactor of two in the range. He further explained that as one rotates around the unit circlein the domain of f, the amount of rotation varies depending on the angle of the point z0.As he stated this, he traced around the unit circle in the domain with his left hand. Hethen moved to the range and explained that the “dial [image vector of Δz] would getspun around to get different circles” as he used his left index finger to represent thevector and moved it in a counter-clockwise direction (Fig. 3). He concluded byremarking that “as we move around to different parts of the circle, the amount oftwisting of the disk varies. So you’re going to get different marked circles, this littledial (rotates pointer finger around in range) is going to rotate as you move throughdifferent circles.”

Judy: Judy began by noting that the Cauchy-Riemann equations plus real differen-tiability imply complex differentiability, then commented, “I don’t think anyone

Fig. 1 Rafael twisting hand

Fig. 2 Rafael rotating and expanding


understands the real differentiable part well.” To explore this, she drew a tangent planefor the real component function u: R2→R and noted the difficulty of conceiving of theerror between u and its linear approximation going to zero “faster than” |z − z0|. Judythen applied her image of the real tangent plane to reason about the geometry containedin the Cauchy-Riemann equations. Noting that for real functions the tangent planes foru and v can behave completely independently, she wondered what relationship theCauchy-Riemann equations imposed between the planes. Writing down the Jacobianmatrix she explained, “So geometrically, I’m thinking of this tangent plane (points toleft plane) and this other tangent plane (points to right plane) as behaving the sameway as multiplication by a complex number” (Fig. 4). As we shall see, Judy has a richimage of this complex multiplication as an amplitwist. However, since she hadintroduced tangent planes for the component functions, she did not shift to thisamplitwist imagery, but rather persisted trying to work out the geometric relationshipbetween the two tangent planes.

Judy noted that, “I haven’t ever done this or visualized the geometry of what itmeans that the derivatives are equivalent to multiplication by a complex number, but it

Fig. 3 Rafael spinning around to get different circles

Fig. 4 Judy’s tangent planes


seems like a fun thing to do,” then spent nine minutes working out the details. Sheinitially became stuck due to confusion from an abuse of notation (using partialderivative notation for the vectors in the tangent plane lying above unit vectors inthose directions, e.g., ux for the vector ⟨1, 0, ux⟩). After the integration portion of theinterview, Judy returned to this reasoning and essentially sorted out that the Cauchy-Riemann equations imply that the tangent plane for v is rotated 90 degrees about avertical axis from the tangent plane for u.

After Judy’s initial attempt to sort out the tangent plane relationship in which shestruggled to identify a geometric interpretation for complex differentiability, we askedher to explain the meaning of the derivative for the function f (z) = z2. She respondedwithout hesitation, “I’m definitely thinking of that one point by point as being a dilationand a rotation,” and rewrote the derivative at the point f 0(z0) = 2z0 as 2 z0j j ei arg z0ð Þ.Drawing a rectangular grid around a point marked z0, she described the action of thelinear map on the grid as a magnification by 2|z0| and rotation by arg(z0) as shown inFig. 5.

Andrew and Luke: In response to Question 1 about the Cauchy-Riemann equations,Andrew replied, “For that, we need a careful definition of the derivative, and [to]explain the Cauchy-Riemann equations, and [to] show that one implies the other.” BothAndrew and Luke indicated that the “easy” or “simple” direction of the if and only ifstatement is deriving the Cauchy-Riemann equations from the complex differentiabilitycondition, which Andrew demonstrated symbolically. Specifically, Andrew started with

the limit definition f 0 zð Þ ¼ limh→0

f zþ hð Þ− f zð Þh

, where h = a + ib, and rewrote f (z) as

u(x, y) + iv(x, y). Indicating that he “would be thinking about that in Cartesian form,” heproceeded to first let b = 0 to obtain ux + ivx, then let a = 0 to obtain vy − iuy. Andrewconcluded that “If you want to have the limit in the horizontal direction or along the realwould be the same as the limit along the imaginary, this forces you to have, … ux = vyand uy = − vx.”

After demonstrating this “necessary” direction of the equivalence, Andrew men-tioned that “the fact that they [the Cauchy-Riemann equations] are sufficient requires alittle bit, that of course I can try to reproduce if you want me to.”We reminded Andrewthat we were looking for a geometric interpretation, and he responded that he alwaysthinks about differentiability of a complex function implies analyticity, which in turn

Fig. 5 Judy’s geometric reasoning of f 0(z) = 2z


implies the function is holomorphic, and thus “does not depend on z.” He acknowl-edged that this was not a geometric explanation, but that this was how he would explainit to students.” He conceded that “I don’t think there is a lot of geometry at that level ofexplanation” but, after a lengthy pause, decided that writing the Cauchy-Riemannequations in polar coordinates “is probably where the geometry is a little more visible.”However, he instead reiterated that he identifies functions as differentiable if they “donot depend on z.”

After Luke stated that the “necessary” direction of the equivalence was “verysimple,” he mentioned that the other direction “is a long and complicated proof, andI don’t think I have a geometric feel for why it’s true.” Like Andrew, Luke insteadchose to discuss something else, which he identified as “off topic slightly.” Specifically,he mentioned that when he teaches multi-variable calculus, he discusses differentiabil-ity of a surface z = f (x, y) in terms of the partial derivatives with respect to x and y. Hecontinued, “We talk about these like these are just like single variable derivatives;you’re just holding one of the variables fixed. Yet, surprisingly, once we have thesetwo, we are always now able to check rates of change in any direction.” Luke arguedthat the Cauchy-Riemann equations seemed similar in the sense that, “due to some typeof smoothness property of these two functions [u(x, y) and v(x, y)], you know some-thing really good happens in a couple of directions, and from that you get that thedifferentiability condition is satisfied.” However, he abruptly ended this explanationand conceded, “I don’t have a good feel for this.”

Andrew and Luke also provided similar responses to Question 2. Andrew began hisdiscussion of the function f (z) = z2 by stating, “the geometry of functions of complexvariables is difficult.” He explained that a careful geometric interpretation of thederivative “depends on which point you choose.” When asked about the geometricmeaning of the derivative of the z2 function, after a long pause, Andrew jokingly said“slope is 2?” Afterwards, he labeled z2 with the variable w, and drew separate z and wplanes. Andrew also characterized the derivative as “the linearization,” and concludedthat, near a point z0, f (z) ≈ f (z0) + 2z0(z − z0). He related this symbolic inscription backto his mapping diagram by illustrating how a neighborhood around z0 = 1 + i getsdilated by “about 3” and rotated. While describing the rotation and dilation, heproduced corresponding gestures that embodied these transformations, as shown inFig. 6.

Andrew used a rectangular grid to illustrate the local linear mapping of nearby z-values and mentioned that one could look at the image of tangent vectors to

Fig. 6 Andrew’s gestures for local dilation (left) and rotation (right)


visualize the rotation and dilation. Andrew added that he personally thinks ofdifferentiation in terms of conformal maps in order to determine local rotationand dilation. Considering another example point z0 = i, he returned to a locallinearization to determine that f (i + ϵ) ≈ − 1 + 2i(ϵ), and concluded that the vectorfrom i to i + ϵ becomes “twice as long and rotated 90 degrees.” He drew this vectorand its image in the domain and codomain, respectively. He then extended hisreasoning to depict the image of a local rectangular grid near z0 = i under f (Fig. 7).

When asked to describe complex differentiation geometrically, Luke also paused forsome time, but remarked that he likes to think back to the real case. He chose toillustrate this idea with the example f (x) = x2, in which case f 0(x) = 2x and f 0(3) = 6.Luke characterized this real-valued derivative as locally scaling points near x = 3 by afactor of 6. Similarly, he explained that “I think you can think of complex derivatives inexactly the same way,” except that complex numbers “have more in them” and thus wehave both scaling and rotation. When mentioning this local scaling and rotation, heused similar gestures to Andrew’s and added that thinking about complex numbermultiplication in polar form helps elucidate why there is a rotation and dilation, whichhe referred to as an “amplitwist.”

Like Andrew, Luke stressed that a derivative describes the local behavior of a functionand similarly illustrated his description using a mapping from the z-plane to the w-plane.Luke also initially drew a rectangular grid in the domain and suggested that one couldfollow this grid into the codomain to see “the bending and the twisting”when zoomed in.He related this zooming in to ascertain the local rotation and dilation to local linearizationin calculus, in the sense that “we zoom in on a curve in Calc I and it looks like a straightline.” Luke paused and clarified that he actually tends to visualize the local neighborhoodin the domain as a disk rather than a rectangular grid. He redrew his mapping so that, neara particular point, “you have a little bicycle wheel” (see Fig. 8), and the derivative willeither expand or contract the wheel, and also “twist the spokes” as a rotation. As he utteredthis description, he rotated his hand over the “bicycle wheel” as illustrated in Fig. 8.

While discussing the function f (z) = z2, Luke clarified that it does not matter where theimage wheel is centered, but rather that the derivative describes how the shape of thewheel will change under the function z2. Hemomentarilymisidentified the local dilation asa constant factor of two because the derivative is f 0(z) = 2z. When questioned about the

Fig. 7 Andrew’s local grid near z0 = i in the domain (on the left) is mapped to a corresponding grid (verticallines and horizontal dashed lines on the right) around f (i) = − 1 that is rotated 90° and dilated by a factor of 2


rotation, he claimed it would not apply because, “for this simple function, it just scales,because it’s two times the original (points to 2z inscription).” Because Luke’s diagramappeared to map a point near z0 = 1 + i, we asked him to compute and discuss f 0(1 + i). Hecorrectly obtained this value 2 + 2i and also evaluated f (1 + i) = 2i. He then drew a newmapping diagram to “see if this makes sense.”

After plotting 1 + i and f (1 + i), Luke maintained that the image of a neighborhoodaround 1 + i “would still look like a bicycle wheel; it would just be twice as big.”Notably, Luke qualified this assertion by stating “I may not be thinking about thisright.” He also asked aloud, “is there a rotation in here too?” Because he initiallydiscussed complex differentiation with respect to its real-valued analog, we directedhim to revisit the meaning of evaluating the derivative f 0(x) at x = 3. This promptedLuke to change his mind and conclude that “it’s not just a scaling factor; it depends onwhat point you’re at!” With this revelation, Luke promptly concluded that there wouldbe a dilation by “about eight” and rotation by π/4. In summarizing this expansion androtation, Luke produced corresponding gestures similar to his previous ones and toAndrew’s gestures. He concluded that “the behavior is different at different points. Butit’s the same kind of thing.” Hence, both Andrew and Luke ultimately incorporatedgeometric descriptions of complex differentiation similar to Needham’s notion of theamplitwist despite initially resisting the idea that it had a geometric interpretation.

Becky: When asked to provide a geometric interpretation for complex differentia-tion, Becky admitted, “I don’t think of that geometrically,” and described this as “a flawin my background.” In response to Question 1, she asked in disbelief, “Geometricsolely?” and emphasized, “I’m really an equation person so I always— I usually attackit first from the equation point of view. So I can do that first, then the other [geometry]might come.” Then similar to Andrew, Becky manipulated the limit definition of f 0(z),approaching along the real and imaginary axes, which she described as “like we’re inCalc III land.” After completing her symbolic proof involving limits, Becky acknowl-edged the question sought a geometric interpretation, but reiterated “I’m an equationperson.” Nevertheless, she consented while predicting she would “talk in circles while Ireason it out.” She discussed how she associates analyticity with the property that “thelevel curves of the u’s and v’s intersect at right angles,” and momentarily considered

Fig. 8 Luke’s gesture for a local rotation of a neighborhood represented as a bicycle wheel


what might follow if she treated this property as her definition of analyticity. Continu-ing in this manner, Becky surmised, “if I’ve got two curves and I think about the anglebetween them and I apply f (z) to them, f (γ1) does whatever and f (γ2) does whatever,but infinitesimally this angle is still the same.” However, she did not see how thiswould help her interpret the Cauchy-Riemann equations, and concluded “I might begoing around in a circle. No, I’m going to do all sorts of horrible computations if I endup doing this” and abandoned the task.

Becky began her response to the z2 question by writing the symbolic inscriptionsf (z) = z2 and f 0(z) = 2z and drawing a mapping from the upper half of the unit disk to afull unit disk. She explained that she wanted to “draw mapping properties” under thissquaring function because “it helps me think about it.” She clarified that she chose toonly use the half-disk because the function is two-to-one. Becky pointed to the 2z andstated, “I’m thinking about this in terms of little infinitesimal somethings.” Sheillustrated this statement by plotting a point in the domain and its corresponding imagepoint in the codomain. Afterwards, she sketched position vectors corresponding tothese two points, and labeled their complex arguments as θ and 2θ, respectively. Beckyreiterated that she was “still thinking in terms of mapping properties so I can turn it intoa discussion about functions.”

This clarification prompted Becky to remember by name Needham’s (1997)amplitwist, and she simultaneously produced a rotation-dilation gesture (Fig. 9). How-ever, she conceded that “every time I read it, I have to redraw it like 10 times before Iget it.” Becky said she looks over Needham’s book several times every time she teachescomplex analysis, “because my students ask for the geometric interpretation [of thecomplex derivative]” and laughed. She shared her typical response to such questions:“You know what, I have a hard time with the geometric interpretation. Here’s what I getout of [Needham’s] book.” Becky attributed some of her difficulty with these geometricinterpretations to her opinion that she has not “internalized it” and therefore “promptlyforget[s] it.”

Fig. 9 Becky’s rotation-dilation gesture for the amplitwist


Becky returned to her mapping diagram, drew a small neighborhood around herpoint in the domain, and repeated the word “infinitesimal” several times, clarifying that“derivatives only talk about what happens near a point.” After a long pause, sheindicated she initially thought that the two in f 0(z) = 2z meant that “infinitesimal circlesgrow,” yet she knew that the squaring function should shrink small circles near theorigin. She confirmed, “things away from the origin, they grow, but things near theorigin shrink” and produced expansion and shrinking gestures to illustrate these tworespective cases (Fig. 10), concluding that her intuition about f 0(z) was incorrect.

When asked to consider the geometry behind a more specific statement that f 0(2) = 4,Becky plotted the points z = 2 and z = 4 on her diagram and pointed out that f and f 0 bothsend real numbers to real numbers. However, she observed that f 0(−2) = − 4, yet f (−2) =4 = f (2); thus, she maintained, “that’s where me thinking about angles in association with[the derivative] is breaking down.” Accordingly, she reiterated that she does not person-ally think about a geometric interpretation of differentiation in the complex setting.

Integration

The responses to the integration task tended to combine various notions from real-valued calculus, but the mathematicians were primarily uncertain about a clear geo-metric interpretation for integration of complex-valued functions. Rafael was the onlymathematician who provided a response. Similar to Needham’s (1997) integrationdescription, Rafael offered a creative and well-developed story about an early exploreron an ocean ship who attempts to plot his route on a map with errors. Rafael mentionedthat the ship’s actual route represents the path of integration that one might break “intolittle line segments … [and] each one of these [segments] is a displacement vector.”When plotting his route on a map, however, each of these displacement vectors isrotated and dilated due to measurement errors. When these modified displacements areall summed they comprise the charted path whose displacement is the integral. Belowwe provide the details to Rafael’s response and a summary of the remaining responses,which were similar to one another.

Fig. 10 Becky’s expansion (left) and shrinking (right) gestures for points far away from, and near, the origin,respectively


Rafael: Visibly excited to answer the question, Rafael noted “This is one of my littlepet topics because I haven’t actually seen this in any textbook, but I’ve finally cookedup some way to explain to myself and my students how a definite integral could bevisualized.” He then started giving possible nautical interpretations of the symboliccomponents of the Riemann sum definition of a contour integral: a function f (z), pathz(t), integrand f (z(t)), partition, sum, and limit. Saying, “Imagine you’re an earlyexplorer in one of these ocean ships and… you want to … trace out your route on amap,” he drew line segments representing small, actual displacements of the ship on itspath, which he labeled Δz. Rafael noted that adding these displacements correspondedto laying them end-to-end, and results in the overall displacement during the trip, thusproviding a real-world interpretation of the path z(t) in the complex plane.

Interpreting an integrand f z tð Þð Þ dzdt requires providing meaning for multiplying theship’s displacements Δz ¼ dz

dt Δt by the complex number f (z) that depends on the ship’slocation z(t). Relying on his geometric interpretation of complex multiplication, Rafaelthus introduced location-dependent errors that multiplicatively scale the measureddistance traveled in a given time and rotate the measured direction of motion. Heuttered, “Imagine that we have a clock that runs fast and slow depending on how warmor cold the temperature is… and as you move to different places on the earth, true northis different from magnetic north.” Representing the velocity error as a dilation, ρ, andthe direction error as an angle, θ, Rafael noted that the measured displacements Δw oneach segment could be obtained from the actual displacements Δz by multiplying bythe complex number f (z) = ρeiθ. Invoking a limit, he then compared the true velocity ofthe ship, dz

dt, with the distorted measured velocity, dwdt ¼ ρeiθ dz

dt ¼ f z tð Þð Þ dzdt : Indescribing the dilation errors, ρ, Rafael held his hands up and drew them slowlyapart, and he twisted his marker over the eiθ portion of the expression as hediscussed the rotation errors, θ.

Rafael returned to the summation and limit aspects of the formal definition to invokea need to additively compound these small errors across the duration of the trip: “If welay these out end-to-end, we could either integrate this or we could break it up into littlepieces and do a Riemann sum approximation so you get a bunch of little Δw’s likethis.” More carefully distinguishing the two approaches, he noted that

When you do lay these end-to-end, what you’re doing— the Riemann sumdefinition would be a summation of Δwk’s and every Δwk is an f (zk)Δzk. Or ifyou want to do a continuous version of it (makes a swooping motion with arm),what you’re doing is defining w(t) to be the integral of f z tð Þð Þ dzdt. And then youintegrate from t = t0 to some variable endpoint T. So, this gives you the exactdescription of your imputed curve.

Finally, invoking Green’s Theorem and the Cauchy-Riemann equations, Rafaelconcluded,

What I think is really cool is if you have a closed loop here (makes real worldpath a closed loop), you can ask yourself, “Are you going to get a closed loophere (points to captain’s chart)?” And generally speaking, you expect the errorsinvolved are gonna mess you up. So that when you plot your route (traces path of


Captain’s chart) it’s not going to close up to where you started (points at startpoint) … the amazing thing is that there are certain situations where you doalways close up (air traces closed loop over Captain’s chart with left index finger)and those turn out to be the situations corresponding to cases where f is analytic...if it is path independent then the relationship between this chart (points to realworld chart) and that one (points to captain’s chart) is that there is some well-defined mapping. For every point z over here (points to real world), there is somewell-defined w over here (draws arrow to the captain’s chart to indicate amapping), we’ll call it big F(z) … the complex anti-derivative of little f. So thatgives you a physical interpretation of the complex anti-derivative as well(Fig. 11).

When asked why he felt compelled to develop such geometric interpretations for hisstudents, Rafael argued,

It is widely accepted that the graphical representation of a real-valued function isa vital tool for calculus instruction. If the geometric motivation and intuitionprovided by such representations were eliminated, calculus would be a lifelesscollection of purely formal procedural operations. The development of skills ingeometrical visualization and geometrical reasoning grow in both difficulty andimportance when students encounter the challenges of multivariable calculus.Unfortunately, this hard-won and valuable visualization toolkit is generallyunder-utilized in the introductory course in complex analysis, which often em-phasizes the purely formal procedural analogies between real and complexanalysis without attempting to clarify the deeper geometrical connections thatlink complex analysis to real-variable calculus. Gauss and Riemann explicitlyrelied on such geometrical intuition and geometrical reasoning. They recastCauchy’s formalistic approach to complex analysis in geometrical terms, as thestudy of conformal mappings.

Remaining Participants: As mentioned above, the other mathematicians did not have animmediate geometric interpretation to complex integration. For example, although

Fig. 11 Rafael’s real world and captain’s chart


Becky immediately responded, “So an integral is just an accumulation,” she thenquestioned herself as she asked, “right?” As she gestured to a small segment betweenher index finger and thumb, she mentioned that she was conceiving “some small tinyincrements,” and compared it to Riemann sums in Calculus I. Discussing areas ofrectangles prompted her to remark, “We’re accumulating stuff. So, this idea of accu-mulation, we’re adding things up. Do I have a picture in my mind? … I’m adding upwhat? … I’m not sure.” Thus, although Becky knew that the integral indicated anaccumulation, she was not able to recognize the geometry behind the integrand. Thismay be attributed to her inability to explain the amplitwist concept, as she commented,“I’m sure it [geometry] makes derivatives easier, but I just haven’t figured out howyet.”

The remaining three mathematicians invoked interpretations of line integrals frommulti-variable calculus, but with significant hesitation and uncertainty. For example,Judy immediately remarked that it was the same as studying line integrals from multi-variable calculus as she reasoned symbolically and remarked, “I teach this where wethink about integrating along the x-axis for f (z) = u(z) + iv(z), so if we integrate alongthe real axis we can write it as the sum, so it’s acting like a real function.” Referring tothe limits of integration, she remarked instead of dividing a and b along the real line,one would “map it to some portion of the curve using alphas and betas and still think ofit as a real integral.” Judy then mentioned that a common and geometrically helpful realline integral application is “work done along the curve … imagine there’s a force fieldand I’m traveling through it.” As she uttered these words she traced a curve in the airwith her hands together and concluded with, “think of f as the force perpendicular to thecurve and see how much work I obtained as I travel along that curve.”

Andrew and Luke each hesitated in their responses. Andrew’s impulse was torespond that he would not think of integration geometrically and added that he wouldhave to think about it. After some pause, he wrote

∫C

uþ ivð Þ dxþ idyð Þ ¼ ∫C udx−vdyð Þ þ i ∫cvdxþ udyð Þ ð1Þ

and indicated that one could easily determine an answer but he didn’t know how tomake geometric sense of it. He attempted to reconcile his symbolic reasoning withcontexts such as work and flux but concluded, “It just means if I would compute theintegral, this is what I would get. I don’t think I have any deep insight. Is there deepinsight?”

Luke’s initial response to the integration question was similar to Andrew’s, as hestated, “Wow, what does that mean? (long pause). Okay, so this is a hard question. Ifyou’re integrating (long pause). This is hard.” Luke plotted two points on the plane,sketched a path between them, and remarked that the difficulty arose because there arevarious paths between the two points. He shifted to symbolic and formal reasoning andcited several theorems. After asking permission to do some “scratch work,” he wrotef (z) = u + iv and dz = dx + i dy and multiplied the two expressions to also derive Eq. 1.Luke articulated that one could split up the integral into the real and imaginary parts,which “represent things like work and flux, depending on which one you are lookingat.” He further remarked that “if you are integrating over a closed loop (traces aclosed loop in air with right index finger) you use Green’s Theorem. Cauchy-Riemann


equations say this one is gone (pointing to real part of integrand) because there is nowork over a closed loop (traces a closed loop in the air with right index finger).” Lukeelaborated how “you get something like Cauchy’s integral formula because this one(points to imaginary part of the integrand) turns out to be independent of the size of theloop.” Luke uttered that the end result would be 2πi and that these theorems yield“some number,” but did not “know what that number means.” Luke concluded, “I don’thave a good feel for it.”

Discussion and Implications

In this study, we found similarities among our participants’ responses, though thesesimilarities present nuanced differences. The first similarity was related to the dynamicsof abstraction; that is, all of the participants began reasoning about the tasks by invokingsymbolic or formal reasoning. When some of the mathematicians later invoked geometricreasoning, they justified and fashioned the details of these concrete interpretations in termsof their formal reasoning. A second similarity regarded our participants’ tendency to fixateon particular features which became barriers to developing their geometric reasoning.Aspinwall et al. (1997) refer to a similar tendency among students as uncontrollablemental imagery. In this section, we summarize these findings in relation to the pertinentliterature and theory and discuss related pedagogical implications.

Summary of Experts’ Reasoning

In discussing complex differentiation, both Rafael and Judy began by stating theoremsor lemmas that provide the necessary and sufficient conditions for complex differen-tiability. They both expressed the importance of understanding the condition of realdifferentiability, though they felt few people understand or value this condition. Thisintroduction allowed Rafael to provide the background that he needed for his con-scious, detailed, and embodied response, wherein each piece appeared purposeful inorder to discuss the geometry behind the amplitwist concept. On the other hand, Judy’sformal reasoning about real differentiability, that she felt she did not understand well,appeared to lead her down an unfamiliar path relating the configuration of the tangentplanes of the component functions to the Cauchy-Riemann equations, which dominatedher reasoning. However, once she engaged in a concrete example, she switched to amapping interpretation between two complex planes and invoked the familiaramplitwist notion.

Luke and Andrew also primarily relied on symbolic reasoning, and it seemed thatdiscussing the specific function f (z) = z2 provoked them to reason geometrically aboutcomplex differentiation. Andrew leveraged his symbolic reasoning by recognizing thatf 0(z) was simply “a linearization,” which he expressed as a local approximation andillustrated as the amplitwist concept. Luke began his symbolic reasoning by comparingthe derivative of f (z) = z2 to the derivative of the function f (x) = x2, which ultimately ledhim to neglect the rotation of the amplitwist concept, but he corrected his error anddiscussed the rotation, dilation, and locality of complex differentiation. Becky, who had aself-professed predilection towards symbolic reasoning, was unable to discuss complexdifferentiation geometrically, although it appeared she had a well-practiced image of


complex multiplication as a rotation and dilation. This was evidenced by her subcon-scious gesturing accompanying her discussion about the derivative of f (z) = z2. Asidefrom Rafael and Judy, the participants primarily invoked descriptions involving rotationand dilation resulting from complex number multiplication rather than matrixmultiplication. This result coincided with the fact that these participants did not fullyanswer the first question about the geometry of the Cauchy-Riemann equations. Theremaining tasks, however, were more amendable to the participants’ descriptions involv-ing linearization, complex number multiplication, and the amplitwist.

Whether the participants’ geometric interpretations were well-practiced or a result oftheir exploration and problem-solving prompted by the interview tasks, their expert useof various symbols and images is instructive. Most of the experts made natural,unconscious, pervasive, and consistent use of geometric diagrams and physical gesturesembodying the rotation and expansion of complex multiplication. These intertwinedembodied and formal meanings were then extended to their interpretation of thederivative, and in Rafael’s case the integral. The participants fluidly switched betweendifferent symbolic and graphical representations as needed to efficiently capture therelationships between the formal and geometric interpretations they were attempting torelate. For example, they all made numerous rapid and fluid selections of rectangularand polar coordinates and use of rectangular and polar grids, and the function images ofthese grids, to best capture key geometric aspects of the local linearity expressed by thederivative. When needed, the participants quickly leveraged the role of partial deriva-tives in the linear map of the derivative and applied the associated notation to facilitatethe emergence of the corresponding symmetries related to the expression of complexmultiplication as a rotation and dilation.

Reasoning geometrically about complex differentiation and integration led some ofour participants to draw parallels to the real setting. For example, Judy, Luke, andAndrew alluded to the real-variable concept of force and work after rewriting Equa-tion 1. Although Becky also attempted to adapt geometric reasoning from real-variablecalculus while addressing the integration task, she could not make sense of what wasaccumulated as a result of such integration. Rafael stood out as the only participant whohad intentionally made an effort to develop a geometric interpretation of complex lineand contour integrals, primarily for his students’ benefit.

The Dynamics of Mathematical Abstraction

As outlined in our theoretical perspective, many theories of the development ofmathematical meaning frame concrete objects, experience, and activity as the sourcematerial for the construction of more generalized, abstract mathematical concepts. Inthis study, we observed experts reason about formal concepts, that were not abstractedfrom more concrete concepts. Rather, they worked to intentionally developed newconcrete interpretations of these formal concepts. These reasoning patterns reflect thehistorical development of complex arithmetic, algebra, and analysis and provide insightinto theories of mathematical abstraction.

Not only did our expert participants exhibit reasoning about mathematics that wasprimarily formal and symbolic, but most of them initially questioned the possibility ofgeometric or other concrete interpretations. Nevertheless, as research mathematiciansusing complex analysis in their own work, they naturally found formal concepts of


complex differentiation and integration to be deeply meaningful in their own right.Three sources appear to contribute to these meanings. In learning complex analysis, theparticipants engaged directly in the formal theory of complex analysis: definitions,theorems, and proofs. Through repetition at various levels, actions within the formaltheory served as the source material for abstraction. Such abstraction of structure oftheir goal-directed activity reflects a central aspect of constructivist learning theoriesdespite its lack of dependence on more concrete ways of reasoning. Additionally, whileour participants learned further mathematics and applied complex analytic tools in theirown fields, these basic tools assumed meanings derived primarily from theiraffordances in these other purposes. In Dewey’s pragmatic characterization of inquiry,the meaning of tools such as the contour integral emerge from their application andevaluation against novel problems. Our experts offered examples for the practical valueof their geometric interpretations such as deriving results about residues, interpretingthe Cauchy Integral Formula in terms of electromagnetic fields, visualizing howpotential functions arising from a partial differential equation correspond to the realworld, understanding a Riemann surface from a spectral analysis of a differentialoperator, or measuring dilatation of quasiconformal mappings. Finally, largely due toa need to convey more concrete meanings to their own students in introductorycomplex analysis courses, they developed concrete ways of interpreting and conveyingthese previously well-worn formal concepts. This pedagogical motivation was largelythe case for several of the mathematicians invoking an interpretation of an amplitwistand Rafael’s motivation behind his navigation interpretation of the contour integral.

As the participants developed concrete interpretations of formal mathematics, theyprovided examples of creating mathematical meaning in a process that distinctlyreflects Dewey’s dialectic image of inquiry. Specifically, in the aggregate, they pro-duced concrete interpretations of mathematical structure that they initially learned andprimarily employ in their research abstractly. The mechanisms of this abstract-to-concrete meaning-making, however, involved a dialectic of rapid interaction betweenembodied, symbolic, and abstract concepts to continually select, monitor, and guide theoverall construction process. For the concept of the amplitwist, we observed all butBecky interpret the meaning of the general and specific derivative values in terms ofthis local imagery of rotation and dilation. Furthermore, the participants’ physicalgestures accompanying this reasoning (typically twisting and expanding a hand overthe linear operator or over the domain or range drawn in a mapping between complexplanes) were automatic, fluid, and repeated. These gestures were deeply embodiedinterpretations of previously well-understood mathematics. Unlike standard interpreta-tions of embodied cognition, these gestures were not derived from extensive experiencein the world of rotating and expanding objects which the mathematicians laterattempted to formalize. Rather, the rich geometry emerged historically and individuallyonly after significant time and experience working with and developing the formalimplications of the generalization of the basic structures from real to complex variables.

Rafael’s creation of a nautical interpretation of a contour integral also represents aconcretization of a concept with abstract origins. In creating this concrete interpretation,he thought deeply about the formal mathematical structure he wanted to reflect, andintentionally selected and molded the storyline to fit it. We can see several aspects ofthis dialectic interplay. For example, the rotation/dilation interpretation of complexmultiplication is mathematically but not conceptually symmetric; one factor is


interpreted as the operator while the other is the argument to the operation. It would bedifficult to concoct a reason for a real-world path to encode dilations in its velocity androtations in its direction. The integrand function of a contour integral, being arbitraryhowever, has more flexibility. As such, Rafael was free to interpret the path andderivative more naturally as a path and velocity in context. In his nautical interpretation,the mathematical nature of the function f (z) then required the amount of dilation androtation to be location-dependent. For example, the errors could not be attributed todefects in the clock or compass (with constant error) or operator inaccuracy (notlocation dependent). Thus, Rafael had to build location-dependence into the story aserrors caused by temperature and declination. These were not arbitrary features of hisstoryline. As in Black’s interactionist theory of metaphor or scientific model, themeanings of both domains, the formal mathematics and a real-world storyline, had torespond to one another in order to cooperate in a successful interpretation.

The mathematicians’ formal reasoning and development of concrete interpretationsof formal mathematics reflect the historical development of complex arithmetic,algebra, and analysis. As discussed earlier, uninterpreted imaginary expressions wereinitially manipulated algebraically as if they were real numbers simply because theyyielded effective results. Formal definitions of derivatives and line integrals wereabductively transferred from the real case to explore their viability in the complex;they yielded surprising new and powerful results, which compelled new meaningderived from this formal potency. As results and familiarity grew, experimentationwith concrete representations led to powerful interpretations such as the Argand plane,Riemann surfaces, and conformal maps.

Due to these observations, we propose that the development of mathematicalmeaning be viewed as emerging from a dialectic between concrete and formal concep-tions rather than flowing in any single direction. While many instances may predom-inantly reflect concrete objects and activity as the source of mathematical thought fromwhich general structure may be derived, it might be equally important to look for casesin which the opposite occurs. More importantly, even in cases where one direction ispredominant, it may be critical to understand the subtle ways in which the opposite issimultaneously operative in a similar dialectic as that observed in our mathematicians.

Uncontrollable Mental Imagery

Aspinwall et al. (1997) define uncontrollable mental imagery as images that tend toobscure an idea rather than to explain. This occurred with Judy’s fixation on tangentplanes for the component functions, interfering with her ability to respond to the tasksusing more familiar reasoning about an amplitwist. Far from serving as a passivedomain on which to interpret her formal reasoning about the Cauchy-Riemann equa-tions, these geometric meanings were accompanied by their own conceptual commit-ments which interactively influenced the process of developing mathematical meaning.Judy’s tangent plane interpretation privileged a separation between the two dimensionsof the output, thus interfering with the imagery of input-output planes necessary tosupport reasoning about the function as a mapping and locally as a rotation and dilation.The burden of constructing new geometric interpretations related to these tangentplanes was high, and thus Judy lost focus on the symbolic expressions, using partialderivative notation to stand for vectors, and consequently became confused. However,


when we asked her about a specific function, she immediately reasoned geometricallyabout the complex derivative as an amplitwist, described the underlying mathematics,and reflexively gestured in ways consistent with that geometric meaning.

Another example of uncontrollable mental imagery occurred when both Becky,Luke, and Andrew only attended to the dilation aspect of the derivative f 0(z) = 2z,which is sufficient in the real setting. The salient appearance of the 2z as an expressionlinear in z while interpreting the derivative as a linear map led them to, instead,misinterpret the derivative as local multiplication by only the factor of two. Accord-ingly, even these three experts’ responses showcase Danenhower’s (2000) notion ofthinking real, doing complex, a phenomenon that he observed in undergraduates. Forinstance, both Danenhower and Troup (2015) found that undergraduates incorrectlyinterpreted the complex derivative as slope. Exacerbating the effect, it appeared thatboth Becky and Luke compartmentalized their global reasoning about the geometry ofthe mapping f (z) = z2 and local interpretations of the derivative f 0(z) = 2z. Thisdisconnect interfered with their ability to discuss the amplitwist concept, especially inBecky’s case. Analogously, Panaoura et al. (2006) found that high school participantscompartmentalized their algebraic and geometric reasoning. Surprisingly then, com-partmentalization, uncontrollable mental imagery, and thinking real - doing complexare not phenomena restricted to only novices.

Future Research

Though several of our participants began their responses to our questions by saying“this is hard” or questioning whether geometric interpretations even exist (especially inthe case of integration), our results demonstrate that geometric meanings of complexdifferentiation and integration are possible and were even prevalent in the mathemati-cians’ reasoning. As Becky mentioned explicitly in her interview, students tend to askabout such geometric interpretations and might expect complex analogs of the geom-etry in real-valued calculus. Though it was not a direct focus of our study, ourparticipants revealed various ways in which they attend to such student inquiries intheir teaching. For instance, Rafael made it clear that he had thought deeply about avery embodied way to discuss integration, one consistent with Needham’s explanationfor the concept. Others (Luke and Becky) simply mentioned that they bring upNeedham’s (1997) amplitwist notion in class. In contrast to Rafael’s deep pedagogicalconsideration about the geometry of complex differentiation, Becky mentioned that sheoften has to reread Needham’s book to recall the finer geometric details and admits tostudents that she herself has a “hard time” with geometric interpretations.

Nevertheless, because the amplitwist concept permeated our mathematician partici-pants’ embodied descriptions of differentiation, instructors wishing to infuse more geom-etry into their complex analysis classes might especially focus on this central concept. Forsuch instructors, recent research demonstrates successful strategies to help students developthe amplitwist concept via DGEs (Troup 2015; Troup et al. 2017). Because the rotation anddilation aspects in the amplitwist are also rooted in complex number multiplication, wehypothesize that instructors might be able to help students embody the abstract notions ofdifferentiation and integration by emphasizing the geometry of complex arithmetic early inthe course. However, as Becky’s response illustrates, geometric knowledge of complexmultiplication does not necessarily guarantee even an expert’s embodiment of complex


differentiation. Accordingly, it seems that care should be taken to explicitly and purposelyconnect these ideas coherently as a continual and productive focus throughout the course;future work could potentially investigate these matters via a teaching experiment. Someparticipants also referred back to real differentiation (Luke and Judy), but got stuck on someof the finer points. It seems likely that students might also have this inclination anddifficulty, especially in light of Danenhower’s (2000) observation of thinking real, doingcomplex. Accordingly, instructors might wish to set the stage by first reconstructing the realderivative as a special case of the amplitwist as outlined at the end of the “historicaldevelopment” section of the literature review. Anecdotally, this approach has shown somepromise in the classroom (Soto-Johnson and Hancock 2018), but could also be the subjectof future research.

Introductory undergraduate complex analysis courses often do not emphasize formalproofs (CUPM, 2015), thus instructors tend to focus on symbolic calculation. However, itseems natural to leverage students’ spontaneous, embodied, and experiential interpreta-tions when teaching complex differentiation and integration. Accordingly, even if math-ematicians might not regularly employ such geometric interpretations of these analyticconcepts in their own work, our participants mentioned that students typically inquireabout this geometry, and there could be pedagogical merit in pursuing such an approach.In particular, by building on students’ geometric intuition associated with visualizingcomplex numbers as vectors, instructors might leverage geometric interpretations ofcomplex number addition and multiplication with the structure of differentiation andintegration, as described at the end of the “historical development” section of the literaturereview. Additionally, Andrew’s, Luke’s, and Rafael’s responses to the differentiation taskinvoked the language of local linearization in order to reconcile the local rotation anddilation of the amplitwist with their symbolic inscriptions such as f (z) − f (z0) ≈ f 0(z0)Δz.Future work could study a pedagogical avenue in which complex analysis instructorsmight draw upon students’ understanding of local linearity in Calculus I to motivate theamplitwist concept. Moreover, because existing research demonstrates that students candevelop the amplitwist concept with DGEs, we intend to create a DGE lab that mimicsRafael’s nautical explanation in hopes of developing students’ geometric interpretation ofcomplex integration. Finally, while this exploratory study had no a priori goal other than toelicit mathematicians’ geometric reasoning, we are curious about what might ensue ifresearchers gave such experts a mathematical task that contextually called for a geometricrepresentation of the complex derivative or integral. Such work might instantiate Dewey’spragmatic characterization of inquiry by elucidating experts’ possible meaning-adjustmentof a tool during the process of authentic problem solving, as opposed towhat wewitnessedin response to the more open-ended (or familiar, in the case of the z2 example) tasksdiscussed in the present paper.

Acknowledgements The authors are grateful to the Noble Foundation for their support of our cross-institutional collaboration.

Compliance with Ethical Standards

Conflict of Interest On behalf of all authors, the corresponding author states that there is no conflict ofinterest.


References

Asiala, M., Brown, A., DeVries, D., Dubinsky, E., Mathews, D., & Thomas, K. (1996). A framework forresearch and curriculum development in undergraduate mathematics education. In [list editors] (Eds.)Research in Collegiate Mathematics Education II. Conference Board of the Mathematical Science Issuesin Mathematics Education Vol. 6 (pp. 1–32). Providence: American Mathematical Society.

Aspinwall, L., Shaw, K. L., & Presmeg, N. C. (1997). Uncontrollable mental imagery: Graphical connectionsbetween a function and its derivative. Educational Studies of Mathematics, 33(3), 301–317.

Black, M. (1962). Models and archetypes. In M. Black (Ed.), Models and Metaphors (pp. 219–243). Ithaca:Cornell University Press.

Breidenbach, D., Dubinsky, E., Hawks, J., & Nichols, D. (1992). Development of the process conception offunction. Educational Studies in Mathematics, 23(3), 247–285.

Brown, J. W., & Churchill, R. V. (2009). Complex variables and applications. Boston: McGraw Hill.Committee on the Undergraduate Program in Mathematics. (2015). Complex analysis. Undergraduate

Programs and Courses in the Mathematical Sciences: CUPM Curriculum Guide, 2015. Washington,DC: Mathematical Association of America. Retrieved June 11, 2016 from http://www2.kenyon.edu/Depts/Math/schumacherc/public_html/Professional/CUPM/2015Guide/Course%20Groups/Complex%20Analysis.pdf

Council of Chief State School Officers & National Governors Association Center for Best Practices (2010).Common core state standards for mathematics. Common Core state standards initiative. Retrieved May29, 2019, from http://www.corestandards.org/assets/CCSSI_Math%20Standards.pdf.

Danenhower, P. (2000). Teaching and learning complex analysis at two-British Columbia universities.Dissertation Abstract International 62(09). (Publication Number 304667901). Retrieved March 5, 2011from ProQuest Dissertations and Theses database.

Danenhower, P. (2006). Introductory complex analysis at two British Columbia universities: The first week –Complex numbers. In F. Hitt, G. Harel, & A. Selden (Eds.), Research in Collegiate MathematicsEducation VI . Conference Board of the Mathematical Science Issues in Mathematics Education (Vol.XX, p. XXXX). Providence: American Mathematical Society.

Dubinsky, E. (1991). Reflective abstraction in advanced mathematical thinking. In D. Tall (Ed.), Advancedmathematical thinking (pp. 95–126). Boston: Kluwer.

Gravemeijer, K., Cobb, P., Bowers, J., & Whitenack, J. (2000). Symbolizing, modeling, and instructionaldesign. In P. Cobb, E. Yackel, & K. McClain (Eds.), Symbolizing and communicating in mathematicsclassrooms: Perspectives on discourse, tools, and instructional design (pp. 225–273). Mahwah: Erlbaumand Associates.

Harel, G. (2013). DNR-based curricula: The case of complex numbers. Journal of Humanistic Mathematics,3(2), 2–61.

Hickman, L. A. (1990). John Dewey's pragmatic technology. Bloomington: Indiana University Press.Karakok, G., Soto-Johnson, H., & Anderson-Dyben, S. (2015). Secondary teachers’ conception of various

forms of complex numbers. Journal of Mathematics Teacher Education, 18(4), 327–351.Lakoff, G., & Núñez, R. (2000). Where mathematics comes from: How the embodied mind brings mathe-

matics into being. New York: Basic Books.Nahin, P. J. (1998). An imaginary tale: The story of

ffiffiffiffiffiffi−1

p. Princeton: Princeton University Press.

Needham, T. (1997). Visual complex analysis. Oxford: Oxford University Press.Nemirovsky, R., Rasmussen, C., Sweeney, G., & Wawro, M. (2012). When the classroom floor becomes the

complex plane: Addition and multiplication as ways of bodily navigation. The Journal of the LearningSciences, 21(2), 287–323.

Paliouras, J. D., & Meadows, D. S. (1990). Complex variables for scientists and engineers (2nd ed.). NewYork: McMillan Publishing Co.

Panaoura, A., Elia, I., Gagatsis, A., & Giatlilis, G. P. (2006). Geometric and algebraic approaches in theconcept of complex numbers. International Journal of Mathematical Education in Science andTechnology, 37(6), 681–706.

Saff, E. B., & Snider, A. D. (2003). Fundamentals of complex analysis with applications to engineering andscience (3rd ed.). New Jersey: Pearson Education, Inc.

Sfard, A. (1992). Operational origins of mathematical objects and the quandary of reification-the case offunction. In G. Harel & E. Dubinsky (Eds.), The Concept of Function: Aspects of Epistemology andPedagogy. Mathematical Association of America Notes, Vol. 25 (pp. 59–84). Washington, DC:Mathematical Association of America.


Soto-Johnson, H. (2014). Visualizing the arithmetic of complex numbers. International Journal of TechnologyIn Mathematics Education, 21(3), 103–114.

Soto-Johnson, H., & Hancock, B. (2018). Research to practice: Developing the amplitwist concept. PRIMUS.https://doi.org/10.1080/10511970.2018.1477889.

Soto-Johnson, H., & Troup, J. (2014). Reasoning on the complex plane via inscriptions and gestures. TheJournal of Mathematical Behavior, 36, 109–125.

Soto-Johnson, H., Hancock, B., & Oehrtman, M. (2016). The interplay between mathematicians’ conceptualand ideational mathematics about continuity of complex-valued functions. International Journal ofResearch in Undergraduate Mathematics Education, 2(3), 362–389.

Tabaghi, S. G., & Sinclair, N. (2013). Using dynamic geometry to explore eigenvectors: The emergence ofdynamic synthetic-geometric thinking. Technology, Knowledge and Learning, 18, 149–164.

Tall, D. O. (2013). How humans learn to think mathematically: Exploring the three worlds of mathematics.Cambridge: Cambridge University Press.

Troup, J. D. S. (2015). Students’ development of geometric reasoning about the derivative of complex-valuedfunctions. Published Doctor of Philosophy dissertation, University of Northern Colorado.

Troup, J., Soto-Johnson, H., Karakok, G., & Diaz, R. (2017). Developing students’ geometric reasoning aboutthe derivative of complex valued functions.Digital Experiences inMathematics Education, 3(3), 173–205.

Von Glasersfeld, E. (1995). Radical constructivism: A way of knowing and learning. London and Washinton:The Falmer Press.

Vygotsky, L. S. (1978). Mind in society: The development of higher psychological processes. Cambridge:Harvard University Press.

Vygotsky, L. (1987). The development of scientific concepts in childhood. In R. W. Rieber & A. S. Carton(Eds.), The collected works of L.S. Vygotsky (Vol. 1, pp. 167–241) (Original work published in 1934).

Wawro, M., Sweeney, G., & Rabin, J. M. (2011). Subspace in linear algebra: Investigating students’ conceptimages and interactions with the formal definition. Educational Studies in Mathematics, 78, 1–19.https://doi.org/10.1007/s10649011-9307-4.

Wertsch, J. V., & Tulviste, P. (1992). L.S. Vygotsky and contemporary developmental psychology.Developmental Psychology, 28(4), 1–10.

Zill, D. G., & Shanahan, P. D. (2013). Complex analysis. Burlington: Jones & Bartlett Publishers.

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Affiliations

Michael Oehrtman1& Hortensia Soto-Johnson2

& Brent Hancock3

Hortensia [email protected]

Brent [email protected]

1 Department of Mathematics, Oklahoma State University, MSCS 426, Stillwater, OK 74074, USA

2 School of Mathematical Sciences, University of Northern Colorado, Ross Hall 2240C, Greeley,CO 80639, USA

3 Mathematics Department, Central Washington University, Samuelson Hall Room 218E, Ellensburg,WA 98926, USA


https://doi.org/10.1080/10511970.2018.1477889

https://doi.org/10.1007/s10649011-9307-4

experts’ construction of mathematical meaning for

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