Journalfor Researchin MathematicsEducation
2004, Vol. 35, No. 3, 164-186
Understanding Primes:
The Role of Representation
Rina Zazkis and PeterLiljedahl
SimonFraser University,Burnaby,Canada
Inthisarticlewe investigate
howpreservice
school(K-7)teachersunderelementary
of
standthe conceptof primenumbers.We describeparticipants'
understanding
todetectfactorsthatinfluencetheirunderstanding.
primesandattempt
Representation
of numberproperties
servesas a lensfortheanalysisof participants'
responses.We
of primalityof numbersis
suggestthatanobstacleto theconceptualunderstanding
thelackof a transparent
fora primenumber.
representation
Keywords:Conceptual
knowledge;Contentknowledge;Numbersense;Preservice
teachereducation;
Teachereducation;
Teacherknowledge;
Whole
Representations;
numbers
Prime numbersare often describedas buildingblocks of naturalnumbers.The
term building blocks can be viewed as a metaphorical interpretation of the
FundamentalTheoremof Arithmetic,which claims thatthe primedecomposition
of a composite numberto primefactorsexists andis unique.Althoughthe uniqueness of primedecompositionpresentsa challenge for many learners,its existence
is the propertythat is usually taken for granted (Zazkis & Campbell, 1996b).
However, it is the existence propertythatis behindthe building-blocksmetaphor,
creating an image of composite numbers being built up multiplicatively from
primes. What are the structureand the propertiesof these buildingblocks?
Therearetwo propertiesin particularthatseem to presenta mysteryto the learner.
One is the existence of infinitely many prime numbers,which entails very large
primes. Anotheris the propertythatprimenumbersare not generatedby a simple
polynomial function. In fact, mathematiciansof differentorigins have struggled
for centuriesto discover a prime numbergenerator.A few successes in this area
have been recorded in the early 1970s (see for example Gandhi's formula in
Ribenboim, 1996), but these developments present considerable mathematical
complexity and are beyond the scope of our investigation.
Althoughthe understandingof elementarynumbertheoryhas been the topic of
a few recent studies (Campbell & Zazkis, 2002), there has not been any study
focusing specifically on primenumbers.On a relatedmatter,Zazkis andCampbell
(1996b) investigatedpreserviceteachers' understandingof prime decomposition
andconcludedthat"if the conceptsof primeandcompositenumbershave not been
Thestudyreported
inthisarticlewassupported
inpartbygrant#410-2002-1120
fromtheSocialSciencesandHumanities
ResearchCouncilof Canada.
Rina Zazkisand Peter Liljedahl
165
adequatelyconstructed,this will likely inhibit any meaningfulconceptualization
of prime decomposition"(p. 217).
Understandingnumbers,ways of representingnumbers,andrelationshipsamong
numbershave been identifiedby the NationalCouncilof Teachersof Mathematics
as importantaspects in the Numberand OperationsStandardfor all grade levels
(NCTM,2000). We believe thatthe significanceof understandingprimesis in each
of these three aspects. As building blocks, primes are crucial in understanding
numbersandmultiplicativerelationshipsamongnumbers.Further,understanding
primesrelies heavily on the representationof numbers,as we illustratein the next
section.
In this article we describe and analyze preservice elementary school (K-7)
teachers'understandingof primenumbersand attemptto detect factorsthatinfluence their understanding.We use representationof numberproperties,and what
can be learnedby consideringrepresentationsof a number,as a lens for the analysis
of participants'responses.We arguethatthe lack of a transparentrepresentation
of prime numbers presents an obstacle for students' understandingof prime
numbers.
REPRESENTATIONOF NUMBER PROPERTIES
A number of research studies in mathematicseducation have addressedthe
issue of representation(e.g., Cuoco, 2001; Goldin& Janvier,1998;Janvier,1987a).
Variousmeaningshave been attributedto the notionof representationandvarious
distinctionsmade. However, only a small partof the work on representationhas
addressedrepresentationof numbers,focusing primarilyon fractionsandrational
numbers(e.g., Lamon, 2001; Lesh, Behr, & Post, 1987), or on negative numbers
(e.g., Goldin & Shteingold, 2001). In the study reportedhere we focused on the
representationsof naturalnumbers.Priorstudies dealing with the representations
of naturalnumbersin the decimal numbersystem focused on the canonicalplacevalue representationand students'understandingof the place-valueconcept (e.g.,
Boulton-Lewis, 1998) as well as the connectionbetween number-namesandtheir
representations(e.g., Fuson, 1990;Miura,2001). Propertiesof naturalnumbersthat
can be expressedby variousrepresentationsremainlargely unexplored.
Lesh, Behr, and Post (1987) introducedthe distinctionbetween transparentand
opaque representationalsystems. According to these researchers,a transparent
has no moreandno less meaningthantherepresentedidea(s)or strucrepresentation
An
ture(s).
opaque representationemphasizes some aspects of the ideas or structuresandde-emphasizesothers.Meira(1998) referredto this distinctionin considerationof instructionaldevices and suggested thattransparencyis not a featureof
an artifactper se but of its use in a specific instructionalactivity. His research
suggested thatacknowledgingtransparencyin use supportslearning.
Borrowingthe terminologyused by Lesh and his colleagues (1987) in drawing
the distinction between transparentand opaque representations, Zazkis and
of numbersandsuggestedthatall such
Gadowsky(2001) focusedon representations
166
Primes
Understanding
representationsare opaque; however, they may have transparentfeatures. For
example,representingthe number784 as 282 emphasizesthatthisis a perfectsquare
but de-emphasizesthe divisibility of this numberby 98. That is, in representing
784 as 282, the propertyof 784 being a perfectsquareis transparentand the property of 784 being divisible by 98 is opaque. Representingthe same number as
13 x 60 + 4 makes it transparentthat the remainderof 784 in division by 13 is 4,
but makes opaque its propertyof being a perfect square.Zazkis and Gadowsky
addressedthe importanceof students'awarenessto propertiesof numbersembedded
in their differentrepresentationsand suggested several instructionalactivities to
promotethis awareness.
In this articlewe extend the representationof specific numbersto the representationof sets of numberspossessing the same propertythroughthe use of algebraic
notation. We say that, for a whole numberk, 17k is a transparentrepresentation
for a multiple of 17, as this propertyis embeddedor "canbe seen" in this form of
the representation.However,it is impossibleto determinewhether17kis a multiple
of 3 by consideringthe representationalone. In this case, we say that the representationis opaque with respectto divisibility by 3.
It often happensthatthe definition for a set of numbersrelies on the existence
of certainrepresentation.For example, a rationalnumberis a numberthatcan be
representedas a/b, where a is an integer and b is a nonzerointeger.Furthermore,
the existence of a certaintransparentrepresentationcan serve as a discriminating
property.For example, a numberis even if and only if it can be representedas 2k
for some whole numberk. A numberis a perfect squareif it can be representedas
k2for some whole numberk. Similarly,we consider5k as a transparentrepresentationof multiplesof 5 and 17k+ 3 as a transparentrepresentationof numbersthat
leave a remainderof 3 when dividedby 17. Findingan appropriaterepresentation
for a given propertyis crucialfor manipulatingrepresentativesof this set.
In learning elementary number theory, consideration of the properties of
oddness/evennessor divisibility of numbersis naturallyaccompaniedby considerationof the propertyof primality.However, there is no transparentrepresentation for prime numbers.We often denote a primenumberby p, but this representationis opaquein everyregard.Primalitycannotbe derivedfromthisrepresentation
in a way that, for example, oddness of a numbercan be derived from the representation 2k + 1. How does this lack of transparentrepresentationinfluence
students'understandingof primes?In orderto addressthis issue we first need to
examinethe significanceof therolesthatrepresentation
playsin acquisitionof mathematicalknowledge.
ROLES OF REPRESENTATIONIN MATHEMATICS
The issue of representationis not new to mathematicseducation.However,it has
recently attracted fresh attention and examination in mathematics education
researchandpractice(Cuoco, 2001; Goldin& Janvier,1998). In the studyreported
here we are interestedin representationsthat unravelpropertiesof numbers.We
RinaZazkisandPeterLiljedahl
167
considerthe standardrepresentationalsystems of symbolsandoperationsin arithmetic and algebra,such as decimal representationof numbersandletterrepresentation of variables. Historically, the achievement of representationwe denote
today as standardwas a lengthy process guided by need and human ingenuity.
However, for today's purposes we consider these representationsas tools for
learning and communicatingmathematics.The following summarydraws on a
variety of ideas presented in prior research (e.g., Cuoco, 2001; Goldin, 1998;
Goldin & Janvier,1998; Goldin & Kaput, 1996; Janvier,1987a; Skemp, 1986).
Toolsfor Manipulationand Communication
Representationis often presentedas a tool for manipulatingobjects. Having a
representationin hand allows an individualto detachhimself or herself from the
meaning of this representationand operate on the symbols alone, making the
manipulations automatic, and returning later to interpretingthe result of the
symbolic manipulation(Skemp, 1986). Further,the natureof the manipulationto
be performedmay influence the choice of representation.For example, multiplication of largenumbersis bettermanipulatedwhen these numbersarerepresented
in Hindu-Arabic,ratherthanin the Roman,numerationsystem. Likewise, for the
purposeof multiplyingcomplex numbers,the polarrepresentationis preferableto
the ordered-pairrepresentationof such numbers.
Communicationis often mentionedas an importantrole of representation(e.g.,
Kaput,1991; Skemp, 1986). As a tool for communicating,representationsserve a
dual purpose: they help in the communication of ideas, and they help in the
communicationbetween individuals. However, a representationitself is just a
string of symbols. Instead, representationcomes to life when learnersmap the
symbols to the mathematicalnotions. This mappingis a two-way streetin that it
enablesthe learnerto communicateideas moreefficientlyandit enablesthe learner
to recognize and interpretwhat ideas are being communicatedby the symbols.
Moreover, availability and awareness of shared representations-be it among
classmates or among researchmathematicians--createsa social milieu for mathematical discourse.
Toolsfor ConceptualUnderstanding
An importantrole of representationin mathematicsis as a tool for thinkingand
gaininginsights(Diezmann& English,2001; Kaput,1987).Researchershavedrawn
strongconnectionsbetween the representationsthat studentsuse and their under& Tabach,2001; Lamon,2001), withunderstanding
connected
standing(Friedlander
to the abilityto apply variousrepresentationsand to choose one thatis appropriate
to the problemsituation.Janvier(1987b) describesunderstandingas a "cumulative
process mainly based upon the capacityof dealing with an 'ever-enriching'set of
representations"(p. 67). Furthermore,representationsare often considered as a
means to form conceptualunderstanding.The ability to move smoothly between
variousrepresentationsof the same concept is seen as an indicationof conceptual
Primes
Understanding
168
andalso as a goal forinstruction(Lesh,Behr,& Post, 1987).Likewise,
understanding
is interresearchby Even(1998) suggeststhatknowledgeof differentrepresentations
twinedwith knowledgeof underlyingnotionsandthe context.
Since actingon mathematicalobjectspromotesthe constructionof corresponding
mentalobjects (Dubinsky,1991; Sfard, 1991), representationaids learnersin their
mental constructions.Representationsare also describedas tools for generalization andabstractionin thatexpressinggeneralitycan be achievedby an appropriate
choice of representation.Moreover, according to Kaput (1991), possessing an
abstractmathematicalconcept"isbetterregardedas a notationallyrichweb of representationsand applications"(p. 61).
METHODOLOGY
Participantsand Setting
The study reportedin this articleis partof ongoing researchon the learningof
elementary topics in number theory by preservice elementary school teachers
(e.g., Zazkis, 1998b; Zazkis, 2000; Zazkis & Campbell, 1996a, 1996b; Zazkis &
Liljedahl,2002). In this ongoing researchwe shadow the learnersin theirmathematics courses, examine theirwrittenwork, and invite volunteersto participatein
clinical interviews. In our analyses of extensive amountsof data, variousthemes
emergethatareworthyof separatefocused attention.In this articlewe presentfragments of this work thatattendto one such theme:primenumbersand representations. These fragmentsdraw on data collected from university studentsenrolled
in a mathematicscourse "Principlesof Mathematicsfor Teachers,"which is a core
course for teachercertificationat the elementary(K-7) level.
The courseinvolved4 hoursperweek of classroominstructionthatwerefollowed
by an open lab tutorial,whereteachingassistantswere availableto addressstudents'
questionsandto engage themin additionalproblemsor activities.Despite the limitations of a large class of 116 students,the instructionwas interactive,and group
work was implementedduringclass time as well as in the preparationof students'
assignments. The usual practice in the course was for the instructorto engage
students in a mathematicalactivity or a problem and then draw conclusions or
discuss concepts with referencesto the activity.For example, studentswere asked
to list all the possible arrangementsof n objects (1 <n <40) in a rectangulararray.
Figure 1 demonstratesfour possible arrangementsfor 6 objects and two possible
arrangementsfor 5 objects.
The notions of prime and composite number,factor, multiple, and divisibility
were discussed and formally defined with a reference to this rectangulararray
activity. Othertopics in the chapteron numbertheory,which were studiedfor the
period of 3-4 weeks, included divisibility rules, prime decomposition and the
FundamentalTheorem of Arithmetic, and least common multiple and greatest
commondivisor.The datawere collectedfrom studentsafterthe completionof this
chapteron elementarynumbertheory.
RinaZazkisandPeterLiljedahl
(a)
169
(b)
for6 objects(a) and5 objects(b).
Figure1. Rectangular
arrayarrangements
Questions
Subjectmatterknowledgeis essentialin learningto teachfor understanding(e.g.,
Ball, 1996).As such,in designingquestionswe consideredwhatpreserviceteaches'
knowledge of prime numberscould and shouldentail. Moreover,we build on the
propertiesof connectednessand basic ideas, as discussed by Ma (1999) as being
among her indicatorsof ProfoundUnderstandingof FundamentalMathematics
(PUFM).
The fact thatprimenumbersarea basic idea, or buildingblock, of numbertheory
was a startingpoint for this research.Further,this basic idea cannotbe approached
in isolation because understandingof a mathematicalconcept presentsa complex
web of relationsand connectionswith otherconcepts. As such, a student'sunderstandingof a primenumberis connectedto the understandingof multiplicativerelationshipsamongnaturalnumbers,thatis, of factors,multiples,compositenumbers,
anddivisibility.Therefore,we believe thatan understandingof a conceptof a prime
numberby an elementaryschool teachershould include at least the following:
a) Awarenessthatany naturalnumbergreaterthan 1 is eitherprimeor composite
and the ability to cite and explain the definitionof a prime number;
b) Understandingthat if a number is representedas a product it is composite
unless the factors are 1 and a prime;and
c) Awarenessthatcompositenumbershave a uniqueprimedecompositionandthat
the numberof primes is infinite (thoughnot necessarily the ability to provide
a formalmathematicalproof for these claims).
For the purposeof this study, we designed the following questions and analyzed
participants'responses to them.
(1) How do you describea primenumber?A compositenumber?Whatis the relationshipbetween prime and composite numbers?
(2) ConsiderF = 151 x 157. Is F a prime number?Circle YES/NO, and explain
your decision.
170
Primes
Understanding
(3) Considerm(2k+ 1), where m and k arewhole numbers.Is this numberprime?
Can it ever be prime?
Our choice of questions relatedprimarilyto the understandinglisted above in
that Question 1 relatedto (a) and Questions2 and 3 relatedto (b), althoughsome
aspects relatedto (c) surfacedin the students'responses.Question 1 was initially
included in the interview as an easy "warm-up"question, intended to create a
conversation and an atmosphereof cooperation. However, the way in which
studentsdescribea conceptprovidesa windowinto theirunderstanding.Therefore,
Question 1 was analyzedfor the ways in which studentsdescribedprimenumbers.
Although students learned the formal definition, we expected to find in their
responses theirways of thinkingaboutprime and composite numbersratherthan,
or in additionto, the citationof the definition.Furthermore,we were interestedin
the relationshipsthatparticipantsidentifybetweenprimeandcompositenumbers.
Question 2 requiresno work because the numberF is representedas a product
of naturalnumbers.We were interestedin seeing to what degree this representation would play a role in the responsesof participants.
Question3 is also concernedwith a numberrepresentedas a product.However,
since the productis representedin algebraicnotation,whetheror not this product
is trivialis not specifiedby constrainingthe values of k andm; thatis, we consider
a productof two numbersto be trivial if one of the factors is 1. In this case, the
productcan be trivialonly if m is 1 or if 2k + 1 is 1. Similarto Question2, the focus
of the analysis of responses to this question was on the participants'attentionto
representation.
Data Collection and Analysis
Question2 soughta writtenresponsefrom 116 students.Questions 1 and3 were
presentedin an interviewsettingto a subsetof 18 volunteers.Because groupwork
was an essential componentof the course, several studentsrequestedto be interviewed together with their team members.The researchersrespondedpositively
to these students'request,whichresultedin the 18 volunteersbeing distributedinto
two groupsof 3, one groupof 2 and 10 individualinterviews.However,the differences between individualresponses and groupinteractionare not analyzedhere.
Calculatorswere availableto studentsduringthe datacollection for this study.
This was consistent with the availabilityof calculatorsduringtheir coursework.
Despitethe unrestrictedavailabilityof calculators,we urgedparticipantsto use them
only where appropriate.
Students' responses were compiled according to the common themes that
emergedwithineachquestion.Furthermore,
buildingon the analysisof the students'
responsesacrossthethreequestions,we suggestedpossibleavenuesfor constructing
primalityas a mental object.
RinaZazkisandPeterLiljedahl
171
RESULTSAND ANALYSIS
Question 1
In Question 1, participantsin the interviewwere asked to describethe meaning
of prime and composite numbersand to describe furtherhow they saw the relationship between them. Prior to this interview, the formal definition of prime
numbers presented to students in the textbook and by the instructorwas this:
Prime numbersare numbersthat have exactly 2 factors. A class discussion took
placeto clarifywhetherthis definitionwas equivalentto whatstudentsrecalledfrom
theirelementaryschool experience;thatis, primenumbersarethose thataredivisible only by 1 anditself. However,the phrase"divisibleonly by 1 anditself' created
ambiguityin the considerationof the primalityof number1. It was pointedout to
the studentsthatby mathematicalconventionthe number1 is not consideredto be
prime, and therefore"havingexactly 2 factors"was a more accurateindicatorof
the propertyof primality.
Meaning of prime and composite numbers. All the interviewed participants
provideda reasonabledescriptionof primeand composite numbers.The wording
of the question-how do you describe a prime number---didnot imply that the
formaldefinitionwas expected. Eight studentsused a variationof a formaldefinition (referringto exactly 2 factorsor 2 distinctfactors),whereasthe other 10 used
a variationof a definition familiarfrom their prior schooling (divisible by 1 and
itself). Two interrelatedthemes emergedin participants'responses- providinga
negativedescriptionandsupplementingthe learneddefinitionwith personalunderstanding.We exemplify these tendencieswith these excerptsfrom the interviews;
all names thatfollow are pseudonyms.
Primenumberscannotbe dividedby anythingotherthan 1 and itself.
Sally:
Compositenumbersarenotprimenumbers.Youwouldhave1 anditselfand
youwouldhaveotherum,whatdo youcallthem..,. multiples?
Tom:
Primesarethosethatcannotbe factored,yea, like cannotbe factoredany
further.Compositenumbersarelike,youcanalwaysfactoroutprimesoutof
them.
Karen:
Primenumberwouldbe a numberthatwouldbe divisibleonlyby 1 anditself,
it meansit wouldn'thaveanyotherfactor,besidesthesetwofactors,1anditself
numbers
aretheonlyfactors.Composite
wouldhaveotherfactorsbesidesthese
two,thesetwoyouwouldalwayshave.
Primenumbers
haveonlytwofactors.Theyarenothavingotherfactorsto be
Jenny:
brokendowninto.Compositenumbershavemorethantwofactors,it means
justonemoreandit makesthenumbercomposite.
Sally and Tom appearedto thinkof primenumbersin termsof propertiesthese
numbersdo not hold in that that the numbers"cannotbe divided"or "cannotbe
factored."We refer to these as negative descriptions.Karen'sresponse contains
the formaldefinition,but it is accompaniedby an additionalnegativeexplanation,
"wouldn'thave any otherfactor."Likewise, in Jenny'sdescriptionthe formaldefinition is cited, but also supplementedwith furtherexplanation of what prime
172
Primes
Understanding
numbers"arenot having."For Karenand Jenny,the definitionitself is not sufficient for creatingandcommunicatingmeaning,andas such,it is supplementedwith
an additionalexplanationthatattemptsto interpretthe definition.Negative descriptions appearedin the responses of 15 out of 18 participantseither in their initial
descriptionof primenumbers(e.g., Sally andTom) or in theiradditionalexplanations (e.g., Karenand Jenny).
Relationship betweenprime and composite numbers.The following excerpts
from two individual interviews with Jenny and Sally are examples of students'
descriptionsof this relationship.
Interviewer:Andhow wouldyou describethe relationship
betweenprimenumbersand
compositenumbers?
A compositenumberis madeupof primenumbers.
Jenny:
Interviewer:Madeup?Inwhatway?
It canbe decomposed
intoprimenumbers,likebrokendowninto.
Jenny:
whatdo youmean?
Interviewer:Whenyou saydecomposed,
it
down
into
smaller
numbersthatwhenputtogethertheycomeup
Jenny:
Breaking
to thebiggernumber.
Interviewer:Puttogether?
Jenny:
Multipliedtogether.
[...]
Interviewer:How wouldyou describethe relationshipbetweenprimeand composite
numbers?
Well,notreally,thereis norelationship,
Sally:
theyareoneortheother.If youhave
a numberandit's primeyouknowthatit's notcomposite.
In Jenny's responsewe recognizeher awarenessof the existence of primedecomposition.In fact, the majorityof the participants(12 out of 18) madesome reference
to primedecomposition.Sally sees primeand compositeprimarilyas disjointsets:
'"Theyareone or the other."This themeappearedin theresponsesof 10 participants,
either exclusively or in conjunction with the allusion to prime decomposition.
Althoughfor this particularquestionSally's view appearsincomplete,it is exactly
the understandingthatis necessaryto deal with Question2 in an elegantmanner.
Question2
Question2 askedstudentsto determinewhetherthenumberF, givenasF= 151x 157,
was prime and to explain theirdecision. Table 1 providesa summaryof students'
responses to Question 2. Since students were asked to circle their decision
(YES/NO), it was unambiguoushow to classify their choice as corrector incorrect. However, it was the students' explanationsof their choice, ratherthan the
choice itself, thatrevealedtheirunderstandingof primes.We identifiedand clustered the justifications studentsprovidedfor their decision aboutthe primalityof
numberF. In Table 1, correct (C) or incorrect(I) refersto the claim itself, rather
thanthejustificationprovidedto supportthe claim. In the sections thatfollow, we
discuss the categoriesof responsesthat studentsgave to Question2.
RinaZazkisandPeterLiljedahl
173
Table1
Summary
of Responsesto Question2
ConsiderF = 151x 157.Is F a primenumber?
CircleYES/NOandexplainyourdecision.
Correctclaims(NO)
74
Justification:
C(1)
C(2)
C(3)
C(4)
C(5)
C(6)
Definitionof primes
Definitionof compositenumber
of algorithm
Application
Lackof closure
Reasoningby example(s)
Other
Incorrect
claims(YES)
Justification:
I(1)
1(2)
1(3)
1(4)
33
19
14
2
3
3
42
Productof primesis prime
of analgorithm
Misapplication
of divisibilityrules
Misapplication
Other
24
8
6
4
Definitionofprime and compositenumbers.As shown in Table 1, a total of 74
out of 116 students(64%)have claimedcorrectlythatF was a composite number.
However, not all the correctanswerswere accompaniedby correctjustifications.
The majorityof the arguments(n = 52) used eitherthe definitionof primenumbers
(n = 33; C(1) in the table) or the complimentarydefinitionof composite numbers
(n = 19; C(2) in the table). The following excerptsexemplify students'responses
in these two categories.
C(1): A primenumberhas exactly 2 factors, 1 and itself. F = 151 x 157 = 23707,
thus it is not prime because 151 and 157 are its factors, as well as 1 and
23707.
Prime
numbersare only divisible by 1 and themselves. F will be divisible
C(1):
not only by 1 and F but also by 151 and 157. ThereforeF is not prime.
C(2): 151 and 157 are both prime. Therefore they are prime factors of their
product,makingtheirproducta composite number.
C(2): Because it is composedof at least 2 factors, 151 and 157, otherthanone and
itself. Thereforeit is composite.
It is interestingto note here that20 out of 52 argumentsclassified as eitherC(1)
or C(2) involved considerationsthatare not essential for answeringthis question;
it is unnecessaryto calculatethe numberF to addressthe question.Furthermore,
the claim that 151 and 157 arebothprime,althoughcorrect,is irrelevantto the case.
In some responsesthis claim was not only mentionedin passing,but also the property of primalityof 151 and 157 was derived after studentsengaged in a considerable amount of calculations, following the learned algorithmfor determining
primalityor some variationof it. This time and energy investment suggests that
primalityof 151 and 157 was importantto these studentsin makingtheirdecision
aboutF.
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Primes
Understanding
Applicationof algorithm.In this category,which we call C(3), the argumentsthat
accompaniedcorrect answers were based on an applicationof an algorithmfor
determiningprime numbers,carriedout for the number23707. As we reported
earlierin this article,studentshad access to calculatorswhile workingon this question. An example of a participant'sresponse follows.
C(3): 423707 = 153.9. Now we check if any of the primenumberslower than 153
divide 23707. 23707 is divisible by 151 and 157 so it is not prime.
In this case, the algorithmwas appliedcorrectlyand lead to a correctconclusion.
However, the need for the algorithmis somewhat troublesomebecause it shows
that these students could not conclude that F was a composite number from
considering its representationas a product.Furthermore,the ability to determine
divisibility of F by 151 only afterchecking by division implies thatthese students
did not have a strong connection between divisibility and multiplication, a
finding that is consistent with findings of prior research (Zazkis & Campbell,
1996a).
Lack of closure. Two studentsbased their argumenton the lack of closure of
primenumbersundermultiplication(labeledC(4) in the table).An exampleof this
argumentfollows:
C(4): F is not primebecause 151 and 157 areprimesandthe set of primenumbers
is not closed undermultiplication.
This response may appearas a ratherimpressiveapplicationof recently acquired
terminology. However, a closer look reveals an incorrect logical implication.
Although the claim thatprimes are not closed undermultiplicationis correct,the
use of this claim to determinethatF is not primeis inappropriate.Lack of closure
means thatthere exists at least one pair of elements in the set such that the result
of a binaryoperationappliedon this pairis not an element in the set. This does not
imply that the result of a binary operationapplied to any two elements does not
belong to the set.
Reasoning by examples. There were 2 studentswho based their argumentson
consideringexamples,whichwe classifiedas C(5) responses.One exampleappears
below:
C(5): 2x3=6,5x7=35,2x5=10,5x3=15
All of these examples illustratethatwhen you multiplya primenumberby
a primenumberyour resultis not a primenumber.Primenumbersare 2, 3,
5, 7, 13, 17, 19 .... None of the productswere primenumbers.
This is a clearillustrationof an empiricalinductiveproofscheme (Harel& Sowder,
1998).Thatis to say, whatconvincesthe studentis a considerationof a finitenumber
of examples, ratherthanF's representationas a product.We note here that all the
examples in this student'sillustrationare examples of familiarsmall primes, and
we will returnto this observationlater.
RinaZazkisandPeterLiljedahl
175
We turnnow to describingstudents'argumentsjustifyingthe incorrectclaim that
F is prime.As shown in Table 1, this claim was madeby 42 out of 116 participants
(36%).
Product ofprimes. Out of these 42 participants,24 claimed thatF was a prime
numberbased on the primalityof 151 and 157. An example of responses that we
called I(1) follows:
I(1): Both 151 and 157 are prime numbers, and 2 prime numbersmultiplied
togetherare going to give anotherprimenumber
Of these 24 participants,8 simply claimed that 151 and 157 were prime,whereas
the other 16 explained in detail how they confirmed the primalityof these two
numbers.The absurdityof this argumentcould be an indication of a profound
psychological inclinationtowardclosure, thattwo of a kind producea thirdof the
same kind.
Misapplicationofan algorithm.In responsesclassified as I(2), 8 studentscalculated the productof 151 and 157 to be 23707 and then tested for primalityof this
product. The approach of 7 of these students was similar to the algorithmic
approachin C(3). The studentsfollowed the algorithmicpartof whatthey learned
in thattheydeterminedthe squarerootof 23707 andthenattemptedto performdivision of 23707 by all primes smallerthatthe squareroot. However, unlike students
applyingthe correctapproachin C(3), they failed to applythe algorithmto its full
extend, consideringonly a partiallist of primenumbers.In some cases, the list of
primes resultedin using familiarprimes up to 19 or 29. In othercases, it is difficult to determinewhich primes were considered(because the list was not explicitly mentioned)to reach the conclusion. An example of an I(2) responseis this:
I(2): 151 x 157 = 23707, =23707= 154
Check all prime numberslower than 154 to see if the numberis primeand
if none of them can divide 23707 then the numberis prime.
One studentused her knowledge of divisibility rules to check primalityof 23707.
She checked all the familiardivisibility rules, thatis, rules for divisibilityby 2, 3,
4, 5, 6, 8, 9, and 10 and concludedthatF was prime.
In the responsesincludedin this categorywe identify an implicit belief thatif a
numberis composite, it must be divisible by a small prime. Similarobservations
were reportedby Zazkis and Campbell(1996b). This belief seems to co-exist with
the explicitly stated awarenessthat in orderto determineprimalityof a number,
all the primessmallerthanits squarerootmustbe checked.Thisbelief also co-exists
with the awareness of existence of "very large" prime numbers and infinitely
many prime numbers.
Misapplicationof divisibility rules. Anotherjustification of the primalityof F
was basedon the misapplicationof divisibilityrules-I(3) in the table.Forexample:
I(3): It is primebecause the last digit in the numberis 7 andthe sum of the digits
176
Primes
Understanding
is the number19. 19 is a primenumberand is not divisible by anythingbut
itself and 1. So F is prime.
Considerationof the sum of the digits is a common test for divisibility by 3 and
by 9. In the aforementionedresponse,this divisibility test was overgeneralizedto
claim that the numberwas prime based on the primalityof the sum of its digits.
The last digit is also mentioned,but not actuallyused in the argument.In several
other cases, studentsjustified the primalityof F by the primalityof its last digit.
Although the numberof studentsmakingthese types of claims is relatively small
(5%), their argumentsconfirmthe observationsof priorresearch,which reported
several similar cases of incorrectgeneralizationof divisibility rules (Zazkis &
Campbell, 1996a).
In categoryI(4), we clusteredotherstudents'argumentsthatwere eitherunique
in this group or cases where the argumentwas incomplete or a student' strategy
was inconclusive basedon the writtenresponse.One of the argumentsin this category requiresfurtherattention:
1(4): The productof two odd numbersis always odd. Thereforemaking F odd
too.
Although only one student demonstratedconfusion between prime and odd in
response to Question2, a similarconfusion surfacedin otherencountersas well,
and we discuss these next.
Question3
In Question3 studentsin a clinical interviewsettingwere asked to commenton
whetheror not a numberrepresentedby m(2k + 1) could be prime.This question
is similar to Question 2 in that it asks the studentsto consider the primalityof a
numberpresentedas a product.However, the explicit use of algebraicnotationto
representthe numberlimited the possibility of applying any of the algorithmic
methods familiarto the participants.Withoutthe option to regress to algorithms,
two mainthemes surfacedin participants'responses:(a) attentionto definition(or
a perceived definition) of prime and composite numbersand (b) the convincing
power of examples.
We note here thatthe participantstendedto change theirminds-at times more
thanonce--during the interview.Those interviewedin a groupsettingwere at times
persuadedby their group membersto change their responses. Othersdid so as a
result of promptingby the interviewer.Thus, ratherthan summarizefrequencies
of occurrences,in the next section we illustratethe two main themes.
Definitionofprimeand compositenumbers.In the followingexcerptfroma group
interview,Dina andSally attemptto convinceDan thatthe numberm(2k+ 1) cannot
be prime.
Primesareonlydivisibleby 1 anditself,so...., yeah,yeah,so if youwerejust
Dina:
to writeit out,likethatcouldbe 1,m,somethingtimesmandthenthenumber
itselfright?It wouldhave4, at least4 yeah.
Rina Zazkisand Peter Liljedahl
177
1...]
Sally:
it in, [pause],becausethisis justsayingm times
Becauseyou'remultiplying
thisright,do youknowwhatI'm saying?It wouldbe m timesthisto be the
finalproductright,so m wouldbe one, andthenthiswouldbe another,and
then1 andwhatevertheanswerof thiswouldbe right,so it couldn'tbe prime
becauseif it wereprimeit wouldjustbethistimesthis,right.You'vegotthese
twothingsinthemiddle,wouldn'tit?Do youknowwhatI mean,doyouknow
whatI'mdoinghere?
Dan:
No.
Sally:
Oh,I don'tknowlike,uh,I don'tknowwhatit's called,butyouknowwhen
youlike,let's sayyou'regoingto say6 is 2 times3...
Factors.Youmeanfactors.
Factors,yeahit's goingto looklikethat...
Becauseif it wereprime,it wouldonlybe multipliedby 1 andit, its factors
wouldbe, primefactorswouldbe 1 anditself.
You'resayingit hasfactors...
It hasotherfactors,so it can'tbe prime...
Dan:
Sally:
Dina:
Dan:
Sally:
Interviewer:And those are?
Sally:
Dina:
m and2k+ 1...
Yeah,becausethat'sthewholepointof primenumbers.
Dina claims in the very beginningof the excerptthat"itwould have 4"-implying
4 factors.Sally communicatesthe same idea by explicitly listing the factorsas 1,
m, 2k + 1, and m(2k + 1). She identifies "These two things in the middle"pointing to m and 2k + 1-as additionalfactors, leading Dina and Sally to the
conclusion that the numbercannotbe prime. This view of primes is incomplete,
ratherthan incorrect.In Question 2, representationof F as a product 151 x 157
assisted participantsin determiningthat F was a composite number.In Question
3, for Dina and Sally, representationof the number as a product presented a
distraction, rather than an asset. This is consistent with Tom's perception in
Question 1 that"primenumberscannotbe factored,"in ignoringthe possibility of
trivialfactorization.
Consideringa prime as a numberthat is only divisible by 1 and itself, it was a
popularchoice in this groupof participantsto assign 1 to m. For Karen,however,
this choice appearedsufficient, and her partnerDebra acceptedthe argument.
Karen:
I wastryingto makethetree,andI thoughtthatI couldn't..,.butif m,if any
numberbesides1, youcanstartmakinga tree.
[...]
Debra:
Karen:
Ok,juststartdoingit becauseI'mnotquitefollowingyou.
factorouta 3 right?
If youhavethisnumberandmis 3, youcanimmediately
Debra:
Okay yeah.
Karen:
factorouta 2, if m is
ButI'm,thatmeansif it's, m is 2 youcanimmediately
factorouta 4. However,if m is 1 youcan't,there'sa
4 youcanimmediately
becausethisis odd,
possibilitythatyoucan't,actuallythere'smaybea certainty
the2k+ 1 is oddthatyoucan'tstartfactoringoutanythingif m is 1.Itfreezes
thisas anoddnumber.So myproposalis, yes it's primeif m is 1.
You'reright.
Debra:
178
Primes
Understanding
Karen'sfirst inclinationis relatedto a familiarprocedureof makinga factortree.
She noticed thatin orderto "factorout"m, the value of m shouldbe "anynumber
besides 1."This leadsherto the conclusionthatm shouldbe 1 in orderfor m(2k+ 1)
to be prime.Further,Karenrecognizes2k + 1 as a representationof an odd number.
However, she could be, at least temporarily,confusing the notions of prime and
odd-a theme thatemerged several times in our investigation.
Following the same view of primes as numbersdivisible only by 1 and itself,
Nora concludedthatthe numberis prime if m is 1 and 2k + 1 is "itself."
Nora:
Primesyoucannotbreak,youcanjustwriteit as 1 anditself,likethenumber
itself,like7 is 1times7, butnothingelse.Thiswillbeprimeif youcan'tbreak
it anyfurther.So if hereyouhaveherem(2k+ 1), to be primem shouldbe 1
andthis[2k+ 1] shouldbe itself.
She furtherdeveloped several examples by solving for k several equationsof the
kind 2k + 1 = <prime>.Paul, on the otherhand, identifiedtwo options:
Paul:
It'spossiblebecausethenit wouldonlybe,like2k+ 1 as onefactor,if 2k+ 1
is a primenumberandm is 1, orof m is a primenumberand2k+ 1 equals1,
thenit wouldalsowork,it wouldalsobe prime.
However, this complete argumentwas providedonly after some promptingfrom
the interviewer. Paul's initial attemptwas to representthe numberm(2k + 1) as
2km + m, which led him astray.Only afterrefocusing on the originalrepresentation as a productwas Paul able to provide a comprehensivesolution.
Convincingpower of examples.Differentperspectiveson the role of examples
in learningmathematicshave been addressedby researchers(Mason,2002; Mason
& Pimm, 1984; Rissland, 1991; Wilson, 1990; Zaslavsky & Peled, 1996). In
particular,the convincing power of examples was acknowledged by Harel and
Sowder (1998) as an empiricalinductive proof scheme. Although it is sufficient
to show an example to supportthe claim thatthe numberm(2k + 1) can be prime,
the way in which studentsfound such an example reveals their understandingof
prime numbers. The following two excerpts from individual interviews with
Sharonand Lisa illustratethe searchfor examples.
Interviewer:Ournextnumberis m(2k+ 1),mis a wholenumber,k is a wholenumber,can
a numberwrittenlikethisbe prime?
Sharon: [Pause.]Um,okay,I'mjustlookingat the 2 hereandI justthoughtif it can
be prime,sayif k wasa 2 and2 x 2 = 4, plus 1 equals5, andif m was 1 then
yeah,it couldwork.
Interviewer:So youshowedthatwhenm = 1 andk = 2 we geta primenumber.Good.Are
thereotherexamples?
differentvaluesfork.]
[Sharonsubstitutes
Sharon: [Pause.]So it worksfor3, doesn'tworkfor4, worksfor5 and6, notfor7, I
don'tknow,doesn'tseemto be a rulehere.
[...]
Interviewer:Let'saska differentquestion.Ournumberis m(2k+ 1).
Lisa:
Okay.
Rina Zazkisand Peter Liljedahl
179
Interviewer:Canit be a primenumber?
Lisa:
I guessit woulddependon whatyouputin fork andform.
Interviewer:Okay.
Lisa:
If youput2 form,I figurebecauseyou'remultiplying
somethingby 2 thenit
meansthatit's divisibleby 2, whichwouldin mybrainmakeit
automatically
notprimeandthenit wouldbe divisibleby morethan...
Interviewer:Okay.
[Lisatriesseveralexamplesatrandom.]
Interviewer:So nowyou'retrying5 fork and6 form, anyinsights?
Lisa:
Well,it's lookingto me likeit's notgoingto get prime.
Both Lisa andSharonturnedto examplesas theirfirstchoice of strategy.However,
Sharon's first choice was a lucky one. Lisa, on the other hand, attemptsseveral
substitutionsat random,concludingthat"it's not going to get prime."Despite the
fact thatSharonanswerscorrectlyandLisa does not, theirunderstandingof primes
appearsto be similar. What makes the difference in their responses is Sharon's
systematic approachin choosing numbers to substitute. However, her system
seems to be guidedby neatnessanddiscipline,ratherthanunderstandingof primes.
Such understandingis demonstrated,for example,by Nora's responsein acknowledging the fact that 2k + 1 should be prime and calculatingk from there.It is also
interestingto note thatLisa immediatelyknew thatsubstituting2 for m will result
in an even numberthatwould definitelynot be a primenumber.However,this realization did not preventher from substituting6 for m minutes later.
In the next excerpt with Wanda,the power of examples is even more explicit.
HavingobservedWanda's quicksubstitutionof numbersto producea primeresult,
the interviewerasks her to consider a different argument,provided by another
student.
Wanda:
5 fork andendedupwitha primenumber.But
Yes,it is possible.I substituted
k doesn'thaveto be prime,justa wholenumber,so I'mjustsaying2 x 4 is 8,
+1is 9 whichis notprime.Sothatjustmeansit canbeprime,butitisn'talways.
Interviewer:Okay,youfoundanexamplewhereit is possible.A studentwashereearlier
andsheclaimedthatthisnumbercannotbe primebecauseit is divisibleby m
andby 2k+ 1 andit is alwaysdivisibleby 1 andby itself,so it hasall these
factorsso it cannotbe prime,whatwouldyoutell her?
Wanda: I wouldtryandshowherexampleswhereit didn'tworkthatway.
Interviewer:Examplesof what?
Wanda: I woulduse theexamplethatI justfoundwhereit doesendupbeingprime.
It's not suchabstractthinking,it's concretenumbers.But thenin termsof
explainingwhyherreasoningdidn'twork,I don'tknowwhatI woulddo.
Wanda claims she would convince her classmate by showing her examples that
contradicther claim. However, she does not attendto the flaw in the reasoning,
explicitly claiming "I don't know what I would do."
In summary,considering primes as numbersthat cannot be representedas a
product ignores the possibility of trivial factorization. Acknowledging trivial
factorizationserved for some studentsas a guide in generatingexamples. Others
180
Primes
Understanding
were led to believe that the numberrepresentedby m(2k + 1) cannotbe prime or
turnedto a guess and check by substitutionstrategy.
UNDERSTANDINGPRIMES:
WHAT HAPPENSWITHOUTREPRESENTATION
The previoussectionspresentedparticipants'responsesto each of the threequestions separately.In this section, we commenton theirresponsesto the threequestions in an integratedmanner.We providea shortsummaryof the resultsandthen
considerpossible approachesto understandingprimality.
As mentionedpreviously,the subsetof participantswho respondedto Question
1 provideda reasonabledescriptionfor a primenumber.It is not surprisingthatall
knew whatprimenumberswere,andit is reasonableto expectthatthe resultswould
have been similar had Question 1 been posed to each of the 116 participants.
However, the responsesto Question 2 and 3 indicatethatthis knowledge is often
not implementedin practice.In fact, only 52 out of 116 students(45%,combining
C(1) and C(2)) implementedtheirknowledge in respondingto Question 2. Thus,
understandingof primes appearedincomplete, inconsistent, fragile, algorithmdriven, and significantlyinfluencedby particularexamples.
Treatingmathematicalconceptsas objectssupportsconstructionof corresponding
mentalobjectsin the mindof students(Dubinsky,1991; Sfard,1991). One possible
way to treat concepts as objects is to involve them as inputs in mathematical
processes, thatis, to act on them or to performoperationson them. In orderto act
on representativesof certainsets of numbers,representation
is an asset.Forexample,
in orderto considerthe sum of two odd numbers,we representthem as 2k + 1 and
2m + 1 andperformthe operationof additionactingon thisrepresentation.
Of course
one can use symbols such as x andy for the two odd numbers,but in this case, no
conclusion aboutthe parityof the sum can be drawnfrom consideringx + y.
Researchersagreethatachievingan objectconceptionof mathematicalconcepts
is challenging(e.g., Sfard, 1991). Supportedby the dataandinformedby the theories of objectconstructionandthe role thatrepresentationmay play in constructing
mental objects, we suggest that the lack of transparentrepresentationfor prime
numberscreatesan obstaclefor actingon them,and,thus,createsan additionaldifficulty for constructing a mental object. In the context of our study, an object
conceptionfor a primenumberis consistentwith responsescategorizedas C(1) and
C(2) to Question2 (see Table 1) andwith Paul's andNora's responsesto Question
3. The majorityof students,however, appearnot to have reachedthis stage. We
next suggest threeinterrelatedapproachesin which studentsconstructtheirunderstandingof prime numbers.
Primalityas an Outcomeof Factoring
One way to constructunderstandingof primalityis as an outcomeof the process
of factoring naturalnumbers.If a numberhas a nontrivialfactor, where trivial
RinaZazkisandPeterLiljedahl
181
factors are 1 and the numberitself, then it is not a prime. If a numberhas only
trivial factors, it is prime. Factoringis closely relatedto the notion of representation,because factoringa numberis synonymouswith representingit as a product
of natural numbers. In this approach,the property of primality is added as a
yes/no propertyassignedto the objectof a naturalnumberthathas been constructed
by learnersat earlier stages as an outcome of generalizationof equivalence and
seriation(Piaget, 1965).
In an attempt to factor a number, how does one look for possible factors?
Sophisticatedandfast algorithmsfor factoringvery largenumbersused in ciphering
arebeyondthe scope of ourinterestshere.The studentsin ourstudyinitiallyrecalled
their school experiencesand associatedthe searchfor factorswith the buildingof
factortrees. Furtherinto the course they were introducedto a standardalgorithm,
where in searchfor nontrivialfactors we check for divisibility by primes smaller
than the squareroot of the numberto be factored.This suggests thatthe property
of primality is assigned to small numbers before it can be assigned to large
numbers.In our study, both correctand incorrectdecisions on the primalityof F
(Question 2) were accompaniedby the algorithm.Out of 116 students,22 (19%,
combiningcategoriesC(3) and I(2); see Table 1) referredto the algorithmin their
response. However, as the data demonstrate,correctapplicationof the algorithm
couldbe quitemeaningless,as it is not neededfordeterminingprimalityof a number
representedin a factoredform.
A few observationsdeserve attentionin discussing the standardalgorithmfor
determiningprimality.One is thatstudentsoften have difficultyin explaining,and
even in reproducingthe explanation,as to why it is sufficientto consideronly divisibility by primes smallerthan the squareroot. This inability to explain leads to a
distrustof the algorithm.We witnessedstudentswho, being familiarwith the algorithm,would check for divisibilityby all the primesup to the number.We observed
studentswho attemptto check for divisibility by all the numbers,not only primes,
up to the squareroot or even up to the numberitself. "Tobe on the safe side" was
the usual reason studentsprovidedto justify these unnecessarycalculations.
It is clear from the datathatthe majorityof students'ideas of factors,multiples,
anddivisibilityarenot well connected.Earlierresearch(Zazkis,2000) demonstrated
that some preserviceteachers do not recognize the claim "B is a factor of A" as
equivalentto the claim of "Ais divisibleby B."This researchalso showedinstances
where, given a numberin its primefactorization,117 = 33 x 132, and a requestto
list all its factors, studentsattemptedto factor 117 by building a factortree, rather
thanto buildfactorsfromits primefactors.In the studyreportedhere a similaridea
is illustratedwith a morestrikingexample:studentsattemptedto performfactoring,
and in some cases failed to do so correctly, when all the factors were explicitly
presentedto them.
Primalityby ObservingExamples
The existence of a transparentrepresentationfor a specific numberpropertycan
help in abstractingand generalizingthatproperty.However, a lack of transparent
182
Primes
Understanding
representationfor primenumbersmay lead studentsto generalizefrom examples.
Of course,generalizingfromexamplestakesplace regardlessof the availablerepresentations;however,with the lack of transparentrepresentation,this approachmay
be preferablefor constructingunderstandingof primes.
Based on examplesfromtheirexperience,the following conclusions aredrawn,
either explicitly or implicitly, by some learners:
* Primenumbersare small;
* Every large number,if composite, is divisible by a small prime number.
This understandingof primesis illustratedexplicitly in the excerptfrom an interview with Tanya,reportedin ZazkisandCampbell(1996b). This student,similarly
to some participantsin our study,claimedthat391 (391 = 17 x 23) was primeafter
dividing it by a few "small"primes.
I guessit's probablymoreexperiencethananything,butit just seemsto me
Tanya:
thatwhenyoufactora numberintoitsprimes,I meanwhenyou'redoingthis,
you'retryingto find the smallest,I meannumbersthatcan no longerbe
brokenintoanythingsmallerasidefrom1 anditself,so that.I guess,it'sjust
thewholeideaof factoringthingsdownintotheirsmallestpartsgivesmethe
ideathatthosepartsarethemselvesgoingto be small(p. 216).
In the studyreportedhere the belief in small primeswas a belief in action. That
is, in their attemptto determineprimality of F (Question 2), eight participants
checked the divisibilityof F only by using a small numberor small primesin order
to draw their conclusion. Although the frequencyof occurrenceof this intuitive
belief among the participantsin this study was small, we have included it here
because of the connectionto priorresearch.
Another property of primes derived by considering examples is that prime
numbersare, for the most part,odd. This observationleads at times to a mathematically incorrectimplication that all prime numbers are odd (Zazkis, 1995).
Furthermore,by confusingthe relationshipbetweenprimeandodd, some students
believe thatodd numbersareprime.This appearsto be a logical misinterpretation
in consideringa necessaryconditionfor primality(for numbersgreaterthan2) as
a sufficient condition.
Primalityby Exclusion: WhatPrimesAre Not
Anotherapproachfor constructingprimalityis by exclusion-by consideringnot
what prime numbersare but what prime numbersare not. This is the idea underlying the famous Sieve of Eratosthenes-the process of "sieving out"composite
numbersfrom a list of naturalnumbersin orderto be left with primes only.
As indicatedby the responsesto Question 1, consideringwhatprimesare not is
a salientfeaturein students'descriptionsof primenumbers.In the wordsof participants, "Primenumbersare those thatcannot be dividedby anything,otherthan 1
anditself" or "Primescannotbe factored."Ratherthan,or in additionto, repeating
a positive definitionsuchas thatprimenumbershave exactly2 factors,the majority
of studentsphrasedtheirdescriptionof primesby attendingto featuresthatprime
Rina Zazkisand Peter Liljedahl
183
numbersdo not possess. In other words, they are not composite numbers.When
the participantswere asked specifically in Question 1 about the relationship
between prime and composite numbers,there was an expectationthat the theme
of prime factorizationwould emerge more frequentlyin participants'responses.
However,for some students,therelationshipbetweenprimeandcompositenumbers
consisted entirely of their membershipin disjoint sets. In students'words, "They
areone or the other,cannotbe both"or "Theyare sortof opposites of each other."
In consideringwhatprimes are not, a number'srepresentationcan be a major
indicator.If primes are numbersthat cannotbe nontriviallyfactored,then representing a numberin a nontriviallyfactored form is definitely evidence that the
numbersarenot prime.However,less thanhalf (52 outof 116,or 45%)of theparticipantsrespondingto Question2 based theirdecision on such a representation.We
believe thatthe reasonfor this finding could be the lack of a strongconnection(at
least in the minds of the majorityof participants)between the factoredformrepresentationof F as 151 x 157 and the ability to identify 151 and 157 as factors or
divisors of F. Polysemy' of the wordfactor may be a confusing factor here, as
numbersthatappearas factors(multiplierandmultiplicand)in a numbersentence
arenot necessarilyfactorsof the product.Anotherpossibleexplanationfor ignoring
couldbe relatedto participants'schoolexperience,in which
the given representation
the decimalrepresentationof numbersprevails,and all the alternativerepresentations are treatedas exercises aimed at uncoveringthe standarddecimal representationof a number.
Although arguing for the importanceof attendingto representation,we also
acknowledgethat for some studentsattentionto representationwas a hindrance.
In respondingto Question3, some studentsclaimedthata numbergiven as a product
cannotbe prime,ignoringthe possibility of a trivialfactorizationas 1 x p.
CONCLUSION
In the studyreportedhere,we investigatedpreserviceelementaryschoolteachers'
understandingof the conceptof primalityof numbers.We have shownthatpossession of the knowledgeof whatprimenumbersareoften does not resultin the implewe havepresented
mentationof thisknowledgein a problemsituation.Furthermore,
three intertwinedways in which participantsunderstandthe concept of prime
number.We have describednumberrepresentationsas transparentor opaquewith
respectto a certainpropertyand arguedthatthe lack of transparentrepresentation
for primalityis an obstaclein constructingthis concept.As Skemp(1986) proposes,
"Makingan idea conscious seems to be closely connectedwith associatingit with
a symbol" (p. 83). However, which symbol can we associate with the idea of
primality?As mentioned above, there is no transparentrepresentationfor prime
The termpolysemyrefersto the propertyof wordshavingdifferentbutrelatedmeanings.The issue
of polysemy is discussed in detail in Zazkis (1998a).
184
Primes
Understanding
numbers.However, there is a transparentrepresentationfor a numberthat is not
prime as a productof two naturalnumbers,neitherof which is equal to 1. While
a lack of representationwas judged to be an obstacle, the availabilityof representation indeed served, for some, as an asset and as a basis for decision making.
Representationis representingsomethingonly if thereis connectionsin students'
mindbetweenthe notionsthatare being representedandthe notionsthey arebeing
representedby.
We agree with Lamon (2001) in her observationthat mathematicseducation
researchis faster in identifying students'deficiencies than in suggesting alternatives. One pedagogicalsuggestionthathas yet to be testedempiricallyis to involve
studentsin the considerationof large numbers.By large we meannumbersthatare
beyond the manipulationabilities of a hand-heldcalculator.A possible variation
of Question 2 would ask studentsto determinewhether,for example, the number
151157is prime.Ourassumptionis thatthe inabilityto rely on calculationswill force
more studentsto attendto the representationthatmakesthe conclusiontransparent.
Over a decade and a half ago, Kaput(1987) claimed that "We give virtuallyno
explicit attentionat any level in mathematicseducationto the relationbetween the
transparencyof certainmathematicalpropertiesor operationsand the representation in which they areencoded"(p. 21). He furthersuggestedthat"naturally,there
is a tight connection between the omissions of the curriculumand the omissions
of the researchcommunity"(p. 21). Ourresearchcomplementsthe recentworkon
representationsin an attemptto make up for this omission.
Researchon whole numbersin mathematicseducationhas traditionallyfocused
on number operations and decimal representations.This implicit tradition is
reflected in a recent National ResearchCouncil reportAdding It Up (Kilpatrick,
Swafford,& Findell,2001) thatdefinedits explicit focus on the notionof a number
and attemptedto synthesize the rich and diverse researchon pre-K-8 mathematical learning.It is not surprisingthatthis reportdoes not mentionconcepts related
to the multiplicativestructureof whole numbers,such as primalityor divisibility.
We consider this to be yet anotherserious omission of researchon the K-8 level
as well as on the teachereducationlevel.
Acknowledgingthese omissions, our studymakes a contributionin two arenas.
We enrichresearchon representationsby focusing on representationof numbers,
distinguishing between transparentand opaque representations,and provide a
theoreticallens for the considerationof the effects thatrepresentationhas on mathematical learning.Further,we extend the researchon preserviceteachers'understandingof concepts in elementarynumbertheoryby focusing on primenumbers.
We hope that this study raises significant concerns that are worthy of further
researchon the understandingof primesin particularand on the understandingof
multiplicativestructuresand relationsof whole numbersin general,as well as on
the role thatrepresentations(or lack thereof)play in the learningof specific mathematicalconcepts.
Rina Zazkisand Peter Liljedahl
185
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Authors
Rina Zazkis, Professor of MathematicsEducation,Faculty of Education,Simon FraserUniversity,
Burnaby,BC V5A 1S6, Canada;zazkis@sfu.ca
Peter Liljedahl, Ph.D. CandidateandGraduateResearchAssistant,Facultyof Education,Simon Fraser
University, Burnaby,BC V5A 1S6, Canada;pgl@sfu.ca