Table Of ContentUpper-division student difficulties with the Dirac delta function
Bethany R. Wilcox and Steven J. Pollock
Department of Physics, University of Colorado, 390 UCB, Boulder, CO 80309
The Dirac delta function is a standard mathematical tool that appears repeatedly in the under-
graduatephysicscurriculuminmultipletopicalareasincludingelectrostatics,andquantummechan-
ics. WhileDiracdeltafunctionsareoftenintroducedinordertosimplifyaproblemmathematically,
students still struggle to manipulate and interpret them. To characterize student difficulties with
thedelta function at theupper-division level, we examined students’responses totraditional exam
questions and a standardized conceptual assessment, and conducted think-aloud interviews. Our
5 analysiswasguidedbyananalyticalframeworkthatfocusesonhowstudentsactivate,construct,ex-
1
ecute,andreflectontheDiracdeltafunctioninthecontextofproblemsolvinginphysics. Here,we
0
focusonstudentdifficultiesusingthedeltafunctiontoexpresschargedistributionsinthecontextof
2
junior-levelelectrostatics. Commonchallengesincluded: invokingthedeltafunctionspontaneously,
n translating a description of a charge distribution into a mathematical expression using delta func-
a tions, integrating 3D or non-Cartesian delta function expressions, and recognizing that the delta
J function can haveunits. Wealso briefly discuss implications of these difficulties for instruction.
9
2 PACSnumbers: 01.40.Fk
]
h I. INTRODUCTION andis typicallynotseenby students until seniorormas-
p ters level analysis courses. Such courses are not often
-
d Researchinto studentdifficulties atthe upper-division taken by physics majors. Here, we will focus almost ex-
e clusively on the δ-function as a tool to describe volume
level is a relatively new but growing area of Physics Ed-
. charge densities in the context of upper-division electro-
s ucation Research [1]. Students in upper-division courses
c statics as this is one of the earliest and perhaps simplest
areaskedtomanipulateincreasinglysophisticatedmath-
si ematicaltoolsastheygrapplewithadvancedphysicscon- uses. Despitethegeneralsenseofmanyfacultyattheau-
y thors’ institution that students will be familiar with the
tent. To explore this dynamic, some of the literature
h δ-functionbythetimetheyreachthejunior-level,wefind
aroundstudentdifficultiesatthislevelhasfocusedonthe
p
thatforsome(arguablymany)studentsjunior-levelelec-
[ use of mathematical tools and techniques during physics
trostatics will be their first exposure to the δ-function.
problem-solving[2]. Forexample,onemathematicaltool
1 Given its many possible uses, we do not claim that the
that appears repeatedly throughout the undergraduate
v researchpresentedherewillspanthespaceofallpossible
physics curriculum is the Dirac delta function (hereafter
0
difficultieswiththeDiracδ-function;however,itwillpro-
9 referred to as simply, the δ-function).
vide a sampling of the kinds of challenges that students
4 The δ-function represents an interesting topic to in-
face when manipulating δ-functions.
7 vestigate student difficulties for several reasons. Despite
0 In this paper we utilize an analytical framework [8]
oftenbeing discussedasapurely abstract,mathematical
. describingtheuseofmathematicaltoolsinphysicsprob-
1 construct, it is rarely used in purely mathematical con-
lemsolvingtostructureourinvestigationandanalysisof
0 texts. In fact, physicists often introduce the δ-function
5 into a problem in order to describe or model a concrete student difficulties with δ-functions (Sec. II). We then
1 present our findings, including common difficulties we
physical system. Additionally, it is the authors’ experi-
: identified in our student population and a brief discus-
v ence that the δ-function is often perceived by experts as
sionofimplicationsforinstruction(Sec.III).Weendwith
i trivialto manipulate, andis oftenintroducedto simplify
X limitations and future work (Sec. IV).
the mathematics ofaproblem. However,we stillobserve
r
a common and persistent student difficulties using the δ-
function. Moreover,whiletheδ-functionhasappearedin
literature around student difficulties with Fourier trans- II. METHODS
forms and measurementin quantum mechanics [3–6], we
are unaware of any existing literature directly targeting Problem solving at the upper-division level is often
student difficulties with the δ-function. considerablylonger and more complex than what is typ-
A physics student may be exposed to the δ-function ical of the introductory level. Making sense of stu-
at several different points in the undergraduate curricu- dents’workaroundtheseupper-divisionproblemscanbe
lum. These include its use as a tool to express volume difficult because students often make multiple mistakes
massorchargedensities,todescribepotentialenergiesor at different stages of a problem, which then propagate
wavefunctions in quantum mechanics, in the context of through their solutions in unpredictable ways. To help
Fouriertransforms,and,formoreadvancedsituations,in manage this complexity, we make use of an analytical
thecontextofGreen’sfunctions. Inmathematicscourses, framework known as ACER (Activation, Construction,
the δ-function is rigorously defined as a distribution [7], Execution,Reflection)to scaffoldouranalysisofstudent
2
difficulties with the δ-function [2, 8]. Inonecase,the studentswerealsoaskedtocalculatethe
integral of the given δ-function expression.
A1-type Prompt:
A. The ACER Framework
Sketch the charge distribution:
The ACER framework organizes the problem-solving
ρ(x,y,z)=cδ(x−1)
process into four general components: activation of
mathematical tools, construction of mathematical mod-
Describe the distribution in words too.
els, execution of the mathematics, and reflection on the
What are the units of the constant, c?
results. These components were identified by studying
expert problem solving [8] and are consistent with both The multiple-response CUE is a research-based, end-
a resources[9] and epistemic framing [10] perspective on of-semester conceptual assessment. Only one question
the nature of knowledge. However,the specific details of on the CUE deals with δ-functions, and it is of the same
how a given mathematical tool is used in upper-division general type as the questions asked on the exams (A1-
problemsolvingisoftenhighlydependentonthecontext type). However, rather than an open-ended prompt, the
in which that tool is being used. For this reason, the CUEasksatwo-part,multiple-choicequestion. Thefirst
ACERframeworkwasdesignedto be operationalizedfor partprovidesthe studentwith the mathematicalexpres-
specific mathematical tools in specific physics contexts. sion that corresponds to the mass density of two point
The operationalization process results in a researcher- masses (i.e., ρ(⇀r)=m1δ(⇀r−⇀r1)+m2δ(⇀r−⇀r2)) and asks
guided outline of key elements in a correctandcomplete whattheintegralofthisexpressionisoverallspace. The
solutionto aparticularproblemorsetofproblems. This secondpartaskswhatthis massdensityrepresentsphys-
process will be discussed in greater detail in Sec. IIC, ically. This questionalongwiththe full CUEinstrument
and additional details about the ACER framework can can be accessed at Ref. [15]. CUE data from CU were
be found in Ref. [8]. collected from four semesters of the E&M 1 course; two
of these were courses for which we also have exam data.
In addition to the CU data, we also collected multiple-
B. Study Context response CUE data from 9 courses at 7 external insti-
tutions. These institutions range from small liberal arts
Data for this study were largely collected from the colleges to large researchuniversities.
first half of a two semester Electricity and Magnetism Think-aloudinterviewswereconductedintwosetsper-
sequence at the University of Colorado Boulder (CU). formed roughly a year apart. Interviewees were paid
Thiscourse,calledE&M1,typicallycoverselectrostatics volunteers who had successfully completed E&M 1 one
and magnetostatics (i.e., chapters 1-6 of Griffiths [11]). to two semesters prior and responded to an email re-
Thestudentpopulationiscomposedofjuniorandsenior- questforresearchparticipants. Allintervieweestookthe
level physics, astrophysics, and engineering physics ma- course during one of the semesters for which we have
jors with a typical class size of 30-60 students. At CU, exam data. Final course grades for interview partici-
E&M1isoftentaughtwithvaryingdegreesofinteractiv- pants ranged from D to A. Interview protocols for both
ity through the use of research-based teaching practices sets of interviews were designed, in part, to probe more
includingpeerinstructionusingclickers[12]andtutorials deeply into difficulties identified in the exam solutions,
[13]. andthus included one ormoreexamstyle (i.e., A1-type)
We collected data from three distinct sources for this questions. In addition, the interview protocols also in-
investigation: student solutions to instructor designed cludedquestionsdesignedtotargetaspectsofthe ACER
questions on traditional midterm exams (N=372), re- framework not accessed by the exams and CUE. These
sponses to one question from the multiple-response Col- questions will be discussed in greater detail in the next
orado Upper-division Electrostatics (CUE) Diagnostic section.
[14](N=146atCU,N=162atexternalinstitutions), and
two sets of think-aloud interviews (N=11).
Examdatawerecollectedfrom7semestersoftheE&M C. Operationalizing ACER
1 course taught by 5 different instructors. Of these, four
were traditional research faculty, and one was a physics The process of operationalizing ACER is discussed in
educationresearcher. Twooftheseinstructors,including detail in Ref. [8]. Briefly, operationalization is accom-
the PER faculty member (SJP), taught the course twice plished through a modified form of task analysis [8, 16]
during data collection. Questions on the exams were de- in which a content expert works through the problems
veloped solely by the instructor for that semester; how- of interest while carefully documenting their steps and
ever, in all cases, the question provided a mathematical mapping these steps onto the general components of
expressionforachargedensityandaskedforadescription the framework. The resulting outline is then shared
and/or sketch of the charge distribution (e.g., A1-type with other contentexperts andrefined until consensusis
prompt below, see Sec. IIC for the naming convention). reached that the key elements of the problem have been
3
accounted for. This expert-guided scheme then serves unitful constant to simply state the total charge (e.g.,
as a preliminary coding structure for analysis of student Q(uniform, spherical shell) = σ∗4πR2).
workand,ifnecessary,isfurtherrefinedtoaccommodate Executionofthemathematics: Thiscomponentof
unanticipated aspects of student problem-solving. the frameworkdeals with elements involvedin executing
To guide our data collection and analysis, we oper- the mathematical operations related to the δ-function.
ationalized ACER for problems involving the use of δ- Because this component deals with actually performing
functions to express the volume charge (or mass) densi- mathematical operations, these elements are specific to
ties of 1, 2, and 3D charge distributions. For example, problems requiring integration of the δ-function.
the volume charge density of a line charge running par- E1: Execute (multivariable) integrals that include one
allel to the z-axis and passing through the point (1,2,0)
or more (potentially multidimensional) δ-functions
can be expressed as ρ(⇀r) = λδ(x − 1)δ(y − 2), where
WhentheresultsoftheintegralsinE1mustbesimplified
λ is a unitful constant representing the charge per unit
for interpretation, Execution would include a second el-
length. Expressingvolumechargedensitiesinthiswayis
ement relating to algebraic manipulation; however,none
often necessarywhen workingwith the differentialforms
oftheintegralsincludedinthisstudyelicitedorrequired
of Maxwell’s Equations and can facilitate working with
significant algebraic manipulation.
the integral forms of both Coulomb’s Law and the Biot-
Reflection on the result: The final component in-
Savart law.
cludes elements related to checking and interpreting as-
The elements ofthe operationalizedACERframework
pects of the solution, including intermediate steps and
for these types of δ-functions problems is described be-
the final result. While many different techniques can be
low. Element codes are for labeling purposes only and
used to reflect on a physics problem, the following three
are not meant to suggest a particular order, nor are all
are particularly common when dealing with δ-functions.
elements always necessary for every problem.
Activation of the tool: The first component of the
R1: Check/determinetheunitsofallrelevantquantities
framework involves identifying δ-functions as the appro-
(e.g., Q, ρ, the unitful constant)
priate mathematical tool. We identified two elements in
the form of cues present in a prompt that are likely to R2: Checkthatthephysicalmeaningoftheunitfulcon-
activate resources associated with δ-functions. stantisconsistentwithitsunitsandtheunitsofall
A1: The question provides an expression for volume other quantities
charge density in terms of δ-functions R3: Verify that the value of the charge in a region is
A2: The question asks for an expression of the volume consistent with expectations
chargedensityofachargedistributionthatincludes While the first two elements are similar, we consider el-
point, line, or surface charges ement R2 to be a distinct and potentially more sophisti-
cated reflection task in that it is seeking consistency be-
We include element A1 because, in electrostatics, δ-
tweenthe student’s physicalinterpretationofthe unitful
functions are often provided explicitly in the problem
constantanditsunits. Thedistinctionbetweenelements
statement, effectively short-circuiting Activation.
R1 and R2 was motivated in part by preliminary anal-
Construction of the model: Elements in this com-
ysis of student work, which demonstrated that the link
ponent are involved in mapping the mathematical ex-
between the units of the constant and its physical inter-
pression for the charge density to a verbal or pictorial
pretationmaynotbeexplicitformanystudents(seeSec.
representation of the charge distribution or vice versa.
IIID).
C1: Relate the shape of the charge distribution to the
The operationalization of the ACER framework for δ-
coordinate system and number of δ-functions
functions revealed that the standard questions typically
C2: Relate the location of the charges with the argu- askedonmidtermexams(A1-type)donotcaptureallas-
ment(s) of the δ-function(s) pects ofproblem-solvingwiththe δ-function. Inparticu-
C3: Establish the need for and/or physical meaning of lar,thistypeofquestionbypassesActivationatanything
more than the most surface level assessment of whether
the unitful constant in front of the δ-function
a student recognize the δ-function. To probe Activation
For problems that also require integration of the δ-
function (e.g., to find total charge from ρ(⇀r)) there are more deeply, we began the first set of think-aloud inter-
views with a question that provided a description of the
additional elements in construction related to setting up
chargedistributionandaskedforamathematicalexpres-
this integral.
sionforthechargedensity(e.g.,A2-typepromptbelow).
C4: Express a differential volume element consistent
This A2-type interview prompt also provides a differ-
with the geometry of the charge distribution
ent perspective on Construction than the A1-type exam
C5: Select limits of integration consistent with the dif- prompt by requiring students to generate a multidimen-
ferential volume element and region of interest sionalδ-functionexpressionratherthanjustinterpreting
Alternatively,forsufficientlysimplechargedistributions, it. However, participants in the first interview set often
one can bypass setting up and computing integrals (el- failed to activate the δ-function in response to the A2-
ements C4 & C5) by using the physical meaning of the type prompt (see Sec. III) and thus never moved on to
4
dealwithConstruction. In orderto targetthe Construc- A1-type questions, the δ-function is given as part of the
tioncomponentmoreclearly,thesecondsetofinterviews prompt, effectively short-circuiting Activation and pro-
directly prompted students to use δ-functions to express viding little information about students ability to recog-
the charge density. nize when the δ-function is appropriate.
Noneoftheinstructor-writtenexamsincludedA2-type
A2-type Prompt:
questions. Instead,thiselementwasspecificallytargeted
What is the volume charge density of an duringthefirstofthetwointerviewsets. Whenpresented
infinitely long, linear charge distribution withthe A2-typepromptshowninSec.IIC,2of5inter-
running parallel to the z-axis and passing viewees spontaneously suggested using δ-functions. The
through the point (1,2,0)? Be sure to remaining three participants all expressed confusion at
define any new symbols you introduce. beingaskedto provideavolumechargedensityforaline
Sketch this charge distribution. of charge. Two of these students attempted to recon-
cile this by defining an arbitrary cylindrical volume, V,
Additionally, only one of the seven exam questions
around the line charge and using ρ = Q/V. This strat-
asked students to integrate an expression containing a
egy, while incorrect, represents a reasonable attempt to
δ-function. Tofurtherprobestudentsabilitytointegrate
makesenseoftheprobleminlieuoftheδ-function. Later
theδ-function(elementE1),bothinterviewprotocolsin-
inthe interview,allfiveofthese students werepresented
cludedquestionsthatpromptedstudentstocalculatethe
withtheδ-functionsexpressionforalinearchargedensity
total charge within a finite region of space. The second
and all but one correctly interpreted this expression as
setofinterviewsalsoendedbyaskingstudentstoperform
describing a line charge. This result suggests that even
asetofcontext-freeintegrationsofvariousδ-functionex-
immediately after completing a junior-level electrostat-
pressions(below)inordertomoreclearlyinvestigatethe
ics course, many students may have difficulty recogniz-
Execution component. These particular integral expres-
ing when the δ-functionis the appropriatemathematical
sionswere eachdesignedto targeta specific difficulty we
toolevenwhentheyareabletoprovideacorrectphysical
anticipated students might have with the procedural as-
interpretation of it.
pects of integrating the δ-function. While all integrals
Threeparticipantsinthesecondinterviewsethadjust
werepresentedwithout a physicalcontext,in the caseof
completed one semester of upper-division quantum me-
integral d) students were also asked if they could come
chanics at the time of the interview. To investigate the
up with a physical situation in which they might set up
context-dependentnatureofstudentActivationofthe δ-
this integral.
function, we began these students’ interviews by asking
Context-free Integral problems: for a mathematical expression for the potential of a fi-
−∞ nite square well in the limit that the well became very
a) R δ(x)dx narrow and very deep. Two of these three participants
∞
spontaneously suggested the δ-function as the appropri-
−∞
atetool. Bothofthesestudents explicitlyfocusedonthe
b) R xδ(x)dx
useofthewords“very narrowandvery deep”justbefore
∞
suggestingthe δ-function. The third student initially at-
10
c) R[aδ(x−1)+bδ(x+2)]dx temptedtowritethepotentialasapiecewisefunctionbut
0 broughtup the δ-function when explicitly told there was
d) RRR aδ(r−r′)r2sin(θ)drdφdθ a more compact way to represent the potential without
usingapiecewisenotation. Successfulactivationoftheδ-
function in the quantum case seemed to be linked to the
III. FINDINGS high degree of similarity between the visual representa-
tion of a deep, narrow potential well and the commonly
used graphical representation of δ(x).
This sectionpresents the identification andanalysisof
commonstudentdifficultieswiththeDiracδ-functionor-
ganizedbycomponentandelementoftheoperationalized
ACER framework (Sec. IIC). B. Construction of the model
The Construction component deals with mapping be-
A. Activation of the tool tween the physics and mathematics of a problem. In the
case of δ-functions, this mapping can be done in two di-
Elements A1 and A2 of the framework represent two rections: from mathematics to physical description, or
different types of of prompts that can cue students’ to from physical description to mathematics. The A1-type
activate resources related to the δ-function. In the case prompts (see Sec. IIB) used on exams, the CUE, and
ofA2-typeprompts,thestudentmustfirstrecognizethat in interviews investigated students’ ability to translate
the δ-function is the appropriate mathematical tool be- a mathematical expression for the charge density into a
foretheycancorrectlyanswerthequestion. However,for physicaldescriptionofthechargedistribution. Aspartof
5
this process, students needed to connect the coordinate
systemandnumberofδ-functionsinthegivenexpression
totheshapeofthechargedistribution(elementC1). For
example, the expression ρ(x,y,z) = cδ(x−1) contains
one Cartesian δ-function and thus represents an infinite
plane of charge. Roughly a quarter of students (23%,
N=87of372)hadanincorrectshapeonexams. Themost
commonerrorwasmisidentifyingvolumechargedensities
that included 1 or 2 δ-functions as point charges (62%,
N=54 of 87).
On the CUE diagnostic administered at the end of
the semester, the fraction of students selecting the in-
correct shape was roughly a third (35%, N=51 of 146).
This trend of selecting the incorrect shape is slightly
morepronouncedinstudentpopulationsbeyondCUwho
have taken the multiple-response CUE (49%, N=79 of
162, 7 institutions) The most common error for this
FIG.1. Examplesofthe‘spike’representation usedby5%of
question was misidentifying the given point mass den-
exam students and 2 of 11 interview students. In this case,
sity (ρ(⇀r) = m1δ3(⇀r −⇀r1) ) as a solid sphere or spher- studentswerepromptedtosketchthechargedistributionde-
ical shell with radius ⇀r1 (76%, N=99 of 130 incorrect scribed byρ(x,y,z)=cδ(x)δ(z−2).
responses, all institutions). This finding is interesting
because, given students’ tendency to misidentify charge
densitiesaspointchargesonexams,wemightexpectthat
ical description of the charge distribution into a mathe-
theywouldbemoresuccessfulatidentifyingpointcharge
matical expression for the charge density. This process
densities. However, we hypothesize that the prevalence
requires students to use the geometry of the charge dis-
of this particular error may have been exacerbated by
tributionto selectanappropriatecoordinatesystemand
the apparent similarity between the expressions for this
number of δ-functions. Of the eight interview partici-
point mass density and that of a spherical shell (i.e.,
pants who attempted to use δ-functions in response to
ρ(⇀r) = m1δ(r − r1)), which are distinguished only by the A2-typeprompt,threewereabletocorrectlyexpress
the presenceofthe cube onthe δ-functionandvectorsin
the line charge density as the product of two 1D Carte-
the argument.
sian δ-functions. Four of the remaining five students in-
In interviews, 9 of 11 participants correctly identified stead attempted to use a single δ-function whose argu-
the shape of one or more charge distributions from the mentwasthedifferencebetweentwovectors,forexample
mathematicalexpressionsforthechargedensity. There- ρ ∝ δ(⇀r −⇀r1) where ⇀r1 = (1,2,z). These students did
maining2studentsbothsketchedthechargedistribution not explicitly acknowledgethis as a 3D δ-function either
on 3D Cartesian axes as a very narrow spike originating verballyor by cubing the δ-function. Three of these stu-
atapointconsistentwiththeargumentoftheδ-function dents also described the line charge as a continuous sum
and extending upwards parallel to one of the axes (e.g., of point charges and integrated their δ-function expres-
seeFig.1). This ‘spike’representationwasalsoseenina sion over all z. This strategy reflects a fundamentally
small number of the exam solutions (5%, N=17 of 311). reasonable physical interpretation of the situation that
The spike drawn by these students is highly reminiscent canbe usedto constructacorrectexpressionforthe line
ofthe1Dgraphicalrepresentationofδ(x)asaninfinitely chargedensity; however,none of these students was able
tall andthin Gaussiandistributionat x=0 that is com- to fully leverage this interpretation to correctly express
monly used when first defining the δ-function. Students the charge density.
whodrawthis‘spike’whensketchingthechargedistribu- Determiningthelocationofthechargedistribution(el-
tion may be attempting to apply this 1D representation ement C2) was not a significant stumbling block for stu-
ofthe δ-functionto a 3Dsketch. The examplesshownin dentsinthisstudy. Noneoftheinterviewstudentsanda
Fig. 1 have a number of additional interesting features. tenth of the exam students (10%, N=39 of 372) drew an
For example, the student in Fig. 1(d) places the spike as incorrect position for the distribution. The most com-
z = 2c rather than z = 2. Additionally, the spikes are mon errors were switching the signs of the coordinates
notalwaysparalleltothesameaxisdespiteallhavingthe (36%, N=14 of 39, e.g., locating the plane described by
sameprompt. These featuresmaywarrantfurther inves- ρ(x,y,z) = cδ(x−1) at x=-1) or having the wrong ori-
tigationinfuturestudies;however,astheywereobserved entation of line or plane distributions (38%, N=15 of
in only a small number of exam solutions and in none of 39). However, all relevant exam questions in this study
the interviews,we will not discuss them in further detail dealt with locating charge distributions described by δ-
here. functionsinCartesiancoordinates,anditispossiblethat
The interviews also included an A2-type prompt (see student difficulties with element C2 would be more sig-
Sec.IIC)toexplorestudents’abilitytotranslateaphys- nificant for other geometries or more abstract notation.
6
The third element in Construction relates to the need cant challenge for our upper-division students.
for a unitful constant in the expression for ρ(⇀r). For
A1-type questions, this constant is provided as part of
the given expression for the charge density, and fully in- C. Execution of the mathematics
terpreting this expression requires recognizing the phys-
ical significance of this constant. For example, in the
TheExecutioncomponentoftheframeworkdealswith
expression ρ(x,y,z) = cδ(x −1), the constant, c, rep-
the procedural aspects of working with mathematical
resents the charge per unit area on the surface of the
tools in physics. The exams provide limited insight into
plane. Roughly a quarter of the exam students (27%,
this component as students were asked to actually cal-
N=70of255)presentedwithanarbitraryconstantspon-
culate an integral of a δ-function in only one semester.
taneously commented on its physical meaning and most
In this case, the students were giventhe expressionfor 3
of these (81%, N=57 of 70) had a correctinterpretation. point charges and asked to calculate R ρ(⇀r)dτ. Roughly
This fraction should be interpreted as a lower bound as
a quarter of the students (27%, N=15 of 56) made sig-
additional students may have recognized the constant’s
nificant mathematical errors related to the δ-function
physical significance but did not explicitly write it down
while executing this integral (element E1). The most
onthe exam. The interviewssuggestthatastudents’in-
common error (73%, N=11 of 15) amounted to a varia-
terpretationoftheconstantcanbefacilitatedorimpeded
tion of equating the integral of the δ-function with the
by their identification of its units. This dynamic will be
integral of the zero point of its vector argument (e.g.,
discussed in greater detail in relation to the Reflection R δ3(⇀r−⇀r1)dτ =R ⇀r1dτ). Similarly,morethana thirdof
component (Sec. IIID).
students(36%,N=53of146atCU,41%,N=66of162at
The A2-type prompts used in interviews, on the other external institutions) selected an incorrect value for the
hand, do not include the unitful constant but instead integral of a point mass density in response to the CUE
require that a student recognize the need for this con- question. The most common error (81%, N=43 of 53 at
stant independently. Two of eight interviewees did not CU,68%N=45of66atexternalinstitutions)wasequat-
spontaneously include a multiplicative constant in their ing the integral of the δ-function with the value (vec-
expression for the charge density. Alternatively, three of tor or magnitude) of ⇀r at which the argument was zero
theremainingparticipantsrecognizedtheneedforamul- (e.g., R δ3(⇀r−⇀r1)dτ = ⇀r1). This issue is different from
tiplicative factor but expressed this factor as ρ(⇀r) (e.g., (though potentially related to) the most common issue
volume charge density = ρ(⇀r)δ(x −1)δ(y −2)). These seen in the exam; however, none of the response options
students appeared to be interpreting the ρ(⇀r) term as on the multiple-choice CUE question matched the out-
representing only the magnitude of the charge density come of the incorrect setup seen commonly on exams.
rather than the magnitude at a specific point in space. Ultimately, these types of errors indicate that students
This also suggests that these students are not treating have correctly recognized that the δ-function somehow
ρ as the quantity defined by convention to be volume picks outthe value⇀r =⇀r1 but apply this reasoninginan
charge density and are instead treating it as the quan- ad hoc fashion in the integration process.
tityconventionallydefinedasλ(i.e.,linechargedensity). The first set of interviews investigated Execution in
Ultimately,fourofthe eightparticipantsincludedamul- thecontextofcalculatingthetotalchargeonauniformly
tiplicativeconstantand articulatedacorrectphysicalin- chargedsphericalshell. Twoofthefiveparticipantswere
terpretationof that constantbefore moving on fromthis notabletocompletethiscalculationandexplicitlystated
question. this was because they could not recall how to integrate
∞
The final two elements in Construction (elements 4 & the δ-function. When asked for the value of R−∞δ(x)dx
5)arenotspecific toquestionsinvolvingδ-functions,but theseparticipantsguesseditwouldbe xor1/x. Extend-
ratherapply to any physics probleminvolvingmultivari- ingthislogicintothreedimensions,thisresponseappears
ableintegration. Theseelementsdealwithexpressingthe consistent with some of the common incorrect responses
differentialvolumeelementandselectinglimitofintegra- ontheCUEandexamquestions(e.g.,R δ3(⇀r−⇀r1)dτ =⇀r1
tion. We havepreviouslyexaminedstudentworkaround or 1/|⇀r1|).
these twoelements in the contextofCoulomb’s law inte- The second interview set (N=6) targeted the first ele-
grals [8] and observed a number of difficulties. However, ment in Execution differently by asking students to per-
onlyasmallnumberofstudents(21%,N=12of56)strug- formthecontext-freeintegrationsshowninSec.IIC.Two
gledwitheithertheCartesiandifferentialvolumeelement students stated that the integral in b) would be equal
or limits of integration in the single semester where the to x without evaluating this expression at x = 0, but
exam question asked for total charge in a finite region none of the six participants had difficulty with the in-
of space. Furthermore, only one of eleven interviewees tegrals in a) or c). Moreover, the 3D δ-functions used
hadcleardifficulties expressingthe differentialchargeel- on the exam and CUE seemed to evoke different, and
ement in spherical coordinates, and one other selected potentiallymorefundamental, difficulties thanthe 1Dδ-
incorrect limits for their Cartesian integral. Thus, set- functions used in interviews. Three of six interviewees
ting up integrals for the relatively simple Cartesian and also evaluated the r integral in part d) in the following
spherical geometries used in this study was not a signifi- way, R δ(r−r′)r2dr = 1r′3, despite correctly executing
3
7
parts a)-c). Two of these students explicitly stated that δ-function is unitless was surprisingly persistent, and in
theeffectoftheδ-functionwasonlytopickoutthevalue some cases interfered with students’ physical interpreta-
′
r = r . This result is consistent with that from the ex- tion of the unitful constant. For example, four of these
ams and CUE, and suggests that a significant fraction ninestudentshadarticulatedacorrectphysicalargument
of our upper-division students have internalized the idea for the units of the constant prior to arguing that the δ-
that the δ-function picks out a particular value of the function was unitless. Two of these students abandoned
variable,butdonotrecognizetheotherimpactsoftheδ- their previous physicalargumentin favor of a unitless δ-
functions on the result of the integral, particularly when function,andtheothertwowereunabletoreconciletheir
dealing with 3D or non-Cartesian δ-functions. intuitionabouttheconstant’sunitswiththeirbeliefthat
the δ-function should be unitless. Ultimately, only 2 of
the9studentswereabletoindependentlyconvincethem-
D. Reflection on the results selvesthattheδ-functionhadunitsof1/m(inthiscase).
The other 7 students were only convinced when specifi-
cally prompted by the interviewer to consider the units
The Reflection component deals with aspects of the
of dx in the expression R δ(x)dx=1.
problem-solving process related to interpreting and
The third element in Reflection deals with using the
checking intermediate steps and the final solution. For
expression for the charge density to calculate the total
the questions used in this study, one of the most pow-
chargewithinafinite regioninorderto ensurethis value
erful tools available for checking and interpreting the
is consistent with expectations. While this is something
various δ-function expressions is looking at units (ele-
thatwesawexpertsdospontaneously,almostnostudents
ments R1 and R2). In particular, for A1-type problems,
(N=3 of 255)explicitly executed such a check on the ex-
looking at the units of the given constant (e.g., the c
ams when not prompted to calculate total charge. Sim-
in ρ(x,y,z) = cδ(x − 1)) can facilitate interpretation
ilarly, only one of the interview students spontaneously
of that constant’s physical meaning. Five of the seven
attempted to calculate total charge in a region to gain
exam prompts explicitly asked students to comment on
confidence in his answer. Interview students were also
the unitsofthegivenconstantandtwo-thirdsofthe stu-
explicitly prompted to calculate totalchargeand to con-
dents (70%, N=178 of 255) responded with the correct
sidertheunitsofthisexpression(elementR1)laterinthe
units (element R1).
interview. This process helped 4 of the 9 students who
Beyondjustcommentingontheunitsofthegivencon-
argued that the δ-function was unitless to realize there
stant, we would like our students to recognize the physi-
was a problem with the units of their expression. These
cal interpretation of this constant (element C3), and en-
findingsunderscorethe potentialvalueofthese reflective
sure that the units and physical interpretation are con-
practices(elementR1-3)bothintermsofcatchingerrors
sistent (element R2). However, it was often difficult to
and facilitating interpretation, but also suggest that our
assess if students did this on the exams, in part because
students rarely engaged in reflection spontaneously.
only a quarter of our students (N=70 of 255) explicitly
commented on the physical interpretation of the unit-
ful constant (element C3, see Sec. IIIB). Most of these
E. Implications for Instruction
students (83%, N=58 of 70) also provided units for the
constant that were consistent with their physical inter-
pretationofitsmeaning. However,athirdofthestudents Thisstudywasnotdesignedtoinvestigatetheimpacts
(32%, N=82 of 255) gave units that were inconsistent of different instructional strategies or curricular materi-
withtheshapetheyidentifiedregardlessofwhetherthey als on the prevalence or persistence of students’ difficul-
commentedonthe constant’sphysicalmeaning. Someof ties with the Dirac δ-function; however, our findings do
these students had the correct shape but incorrect units suggest several implications for teaching or using the δ-
(40%, N=33 of 82), and others had the incorrect shape function at the upper-division level. For example, in-
but correct units (30%, N=25 of 82). For an expert, structors should be aware that students are unlikely to
the units of the constant, its physical meaning, and the encounter the δ-function in their required mathematics
shape of the charge distribution are tightly linked; how- courses,andthatitmayormaynotbecoveredinamath
ever,theseresultssuggestthatthisrelationshipmaystill methods course run within a Physics department. Thus,
be developing for the students. the assumption that all students in a junior-level course
Interviews offer additional insight into the connection willbeawareoftheδ-functionanditspropertiesmaynot
betweentheunitsandphysicalinterpretationoftheunit- be justified. Students’ struggles with the procedural as-
ful constant. When prompted to comment on units, 9 of pectsofintegratingmorecomplexδ-functionexpressions
11 participants explicitly argued (incorrectly) that the may be one manifestation of this lack of sufficient prior
δ-function is unitless and thus, regardless of the geome- experiencewiththeδ-function. Thesestudentsmayben-
try of the charge distribution, the units of the constant efit from additionalopportunities to practice integrating
mustbe C/m3. Thisargumentwasoftenjustifiedbythe both 1D and 3D delta functions and in multiple coordi-
statement that the δ-function was ‘just a mathematical nate systems.
thing’ and thus did not have units. The belief that the Additionally, canonical δ-functions questions rarely, if
8
ever,requireastudenttoconsiderwhen theδ-functionis arise from its uses in contexts outside of electrostatics.
anappropriatetool. Questionsthatdescribeachargedis- Forexample,ourstudentsoftenstruggledtoactivatethe
tributionandaskforanexpressionforthechargedensity δ-functions as the appropriate mathematical tool when
canprovideabaselineassessmentofstudentabilitytoac- not explicitly prompted. As our upper-division students
tivate the δ-functionwhennotpromptedexplicitly. This progress forward through the undergraduate and gradu-
typeofquestionalsoaddressesanotherfinding,thatcon- ate curriculum, it becomes increasingly important that
structing a mathematical expression for the charge den- they be able to recognize situations in which particular
sityisadistinctandpotentiallymorechallengingtaskfor toolswillbeuseful. Additionally,wefoundthatstudents
our students than interpreting that same expression. As have a greater degree of difficulty translating a verbal
the former task is arguablythe more authentic, students description of a charge distribution into a mathematical
may benefit from additional opportunities to construct formula for volume charge density than the reverse pro-
variousδ-functionexpressionsinmultiplecoordinatesys- cess. While interpreting a mathematical formula for the
tems. charge density is a valuable skill for our physics majors,
The belief that the δ-function is unitless was a sur- theabilitytoconstructthatsameformulafromscratchis
prisingly prevalent and persistent idea. This belief may potentially an even more valuable skill that our physics
be exacerbated by presenting the δ-function as a purely majors are likely to use in the future.
abstract mathematical construct. Moreover, the idea of Ourstudentsalsoencountereddifficultieswiththepro-
a unitless δ-function can interfere with students’ inter- cedural aspects of integrating 3D and/or non-Cartesian
pretation of the unitful constant. To facilitate student δ-functions despite often recognizing that the δ-function
reflection on problems involving the δ-function, specific picksoutthevalueoftheintegralatasinglepoint. These
emphasis should be placed not only on the fact that the difficultiesmanifestbothinsolvingintegralsembeddedin
δ-function can have units, but also on how to determine aphysicscontextandthosepresentedinapurelymathe-
them based on the argument of the δ-function. matical context. Finally, our students demonstrated sig-
nificantdifficultydeterminingtheunitsoftheδ-function,
thus limiting their ability to interpret or check their ex-
IV. CONCLUSIONS pressions for the charge density.
Additional work is needed to identify student difficul-
This paper contributesto the limited body of research ties when utilizing the δ-function in other contexts such
on student difficulties with the Dirac δ-function by pre- as quantum mechanics, Fourier transforms, and Green’s
senting an application of the ACER framework to guide functions. Such investigations could also provide a lon-
analysisofstudentproblem-solvingwiththeδ-functionin gitudinal perspective on the growth of student under-
thecontextofexpressingchargedensitiesmathematically standing over time, allowing researchers and instructors
in junior-levelelectrostatics. The ACER frameworkpro- to focus their efforts on addressingthose difficulties that
videdanorganizingstructureforouranalysisthathelped are most common and most persistent throughout the
us identify nodes in students’ work where key difficul- physics curriculum.
ties appeared. Italsoinformedthe developmentofinter-
view protocols that targeted aspects of student problem
solving not accessed by traditional exams, particularly
ACKNOWLEDGMENTS
around activating the δ-function as the appropriate tool
andexecutingintegralsofvariousδ-functionexpressions.
Our upper-division students encountered a number of ParticularthankstothePER@CgroupandMarcosD.
issueswhenusing/interpretingtheδ-function. Thesedif- Caballero for their feedback. This work was funded by
ficulties represent a subset of students’ difficulties with NSF-CCLI Grant DUE-1023028 and an NSF Graduate
the δ-function and may not include issues that might Research Fellowship under Grant No. DGE 1144083.
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