Validations of Proofs Considered as Texts: Can Undergraduates Tell Whether an Argument
Proves a Theorem?
Author(s): Annie Selden and John Selden
Reviewed work(s):
Source: Journal for Research in Mathematics Education, Vol. 34, No. 1 (Jan., 2003), pp. 4-36
Published by: National Council of Teachers of Mathematics
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Journalfor Researchin MathematicsEducation
2003, Vol. 34, No. 1,4-36
Validations of Proofs Considered
as Texts: Can Undergraduates
Tell Whether an Argument
Proves a Theorem?
Annie Selden, TennesseeTechnological University
John Selden, MathematicsEducationResources Company
Thisarticlereportson an exploratory
and
studyof the waythateightmathematics
secondaryeducationmathematics
majorsreadandreflectedon fourstudent-generto be proofsof a singletheorem.Theresultssuggestthat
atedarguments
purported
suchundergraduates
tendto focuson surfacefeaturesof arguments
andthattheir
whetherarguments
areproofsis verylimited-perhapsmoreso
abilityto determine
thaneithertheyortheirinstructors
recognize.Thearticlebeginsby discussingarguments(purported
that
proofs)regardedas textsandvalidationsof thosearguments,
is, reflectionsof individuals
checkingwhethersucharguments
reallyareproofsof
theorems.Itrelatesthemathematics
researchcommunity's
viewsof proofsandtheir
validations
to ideasfromreadingcomprehension
andliterarytheory.Then,a detailed
is givenandtheeightstudents'valianalysisof thefourstudent-generated
arguments
dationsof themareanalyzed.
Key Words:Advancedmathematical
thinking;College/university;
Languageand
mathematics;
Proof;Reasoning;
Writing/communication.
Proofs, that is, arguments'thatprove theorems,have been viewed from a wide
variety of perspectives.Previous studies have consideredthe structureof proofs
(Leron, 1983), theirexplanatorypower (Hanna,1989), errorsand misconceptions
in college students'proofs (Selden & Selden, 1987), the logic involved (Dubinsky
& Yiparaki,2000; Selden & Selden, 1995), universitystudents'"proofschemes"
(Harel & Sowder, 1998), and generic proofs (Rowland, 2002). In addition, one
mightdescribethe genreof proofsor discusstheirrhetoric.Considerablymorecould
also be done in examiningthe process of proof construction.
1We areusing argumentthe way it is commonlyused in the English-speakingmathematicalcommunity (Bagchi & Wells, 1998), whereproofis often reservedfor a correctargumentconsistingof a rather
special kind of deductivereasoningthatwarrantsthe truthof a theorem(dimonstration,in French).In
thatcontext, and in this article,argumentrefersto the same kind of deductivereasoning,except thatit
might not be correct.We arenot referringto the moregeneralkindsof argumentsused outsideof mathematics or to argumentation,for example,to discourseintendedto persuadethatmightincludealluding
to experiences of the intendedaudience(see Perelman& Olbrechts-Tyteca,1969 or Toulmin, 1958).
Thisstudywas supported
in partby a FacultyResearchGrantfromTennessee
Technological
University.
Annie Selden and John Selden
5
Herewe focus on proofs as texts thatestablishthe truthof theoremsandon readings of, andreflectionson, proofsto determinetheircorrectness.We call suchreadings and the mentalprocesses associatedwith them validationsofproofs (Selden
& Selden, 1995). A validationis often much longer and more complex than the
writtenproof and may be difficult to observe because not all of it is conscious.
Moreover,even its conscious partmay be conductedsilently using inner speech
and vision. Validationcan include asking and answeringquestions, assentingto
claims,constructingsubproofs,rememberingor findingandinterpretingothertheorems anddefinitions,complyingwith instructions(e.g., to consideror name something),andconscious (butprobablynonverbal)feelings of rightnessor wrongness.
Proof validation can also include the production of a new text-a validatorconstructedmodificationof the writtenargument- thatmight include additional
calculations,expansionsof definitions,or constructionsof subproofs.Towardthe
end of a validation,in an effort to capturethe essence of the argumentin a single
trainof thought,contractionsof the argumentmight be undertaken.An annotated
transcriptof a sample (hypothetical)validationof a calculus proof can be found
in AppendixI of Selden and Selden (1995, pp. 143-147).
In the sections that follow, we discuss a theoreticalperspectiveon arguments,
proofs, and validationsand compareit with various20th-centurytheoriesof texts
and reading.We then give a detailedtextualanalysis of a single theoremand four
brief student-generatedargumentspurportingto prove it. This analysis provides
groundingfor the previous theoreticaldiscussion, as well as backgroundfor the
subsequentdescriptionand analysis of eight undergraduatestudents'validations
of the samefourarguments.In the concludingsections,we offer some implications
for teachingand pose some questionsfor futureresearch.
VALIDATIONAND OTHERKINDS OF READING
Ourview of the natureof validationis reminiscentof late 20th-centuryideasabout
readingfoundin workson readingcomprehensionandliterarytheoryandcriticism
(as describedin Bogdan & Straw, 1990 and Pearson,2000). In these, the onus for
the constructionof meaningis on the readerratherthanon the authoror the written
text. Each readeris seen as constructinghis or her own story from personalbackground together with the written text. Different readers, or the same reader at
different times, are seen as benefiting differently from a single written text.
Validatorsof proofs, too, may benefit in differing ways accordingto their backgrounds.Forexample, some may develop an understandingof a theoremthatgoes
well beyondjust knowing it is true, and Hanna(1989) has pointed out that some
proofs may be betterfor this than others. Also validationcan sometimes involve
emotionally intense reflection resultingin knowledge construction,for example,
by forging new links between one's ideas.
At the beginning of the 20th century,theorists in reading comprehensionand
literary criticism regarded writing as mainly a matter of communication-the
authorwas consideredtheultimatearbiterof the meaningof a writtentext.To under-
6
Validations
as Texts
of ProofsConsidered
standa text, one studiedthe life and times of the author.By midcentury,the locus
of meaninghadshiftedto the text itself,2andby the end of the century,the construction of meaningwas viewed as residingwith the reader.The idea thatthe meaning
is in the text has been articulatedby Olson (1977); it has been referredto as the
"doctrineof autonomoustexts"andwas critiquedby Nystrand,Doyle, andHimley
(1986), who includedan examplethatmay shed some light on the uniquecharacter
of proofs. They pointedout thatlegal contractsaredocumentsexplicitly meantto
be autonomous and independentof context, but nevertheless such documents
cause disputesthatcourtsareaskedto settleby calling in expertsto clarifythe original terminology and context. This example supportsthe idea that independence
of context is inherentlyunachievable.In contrast,such situationsdo not occurwith
proofs; one does not call in experts to find out what an authormeant by, say,
"compact"to establishthe truthof a theorem.
Like midcentury structuralcritics, mathematicians3seem to treat a proof as
being independentfrom its author.If something,perhapsthe author'sreputation,
should suggest that a proof is probablycorrect, mathematiciansmay attemptto
suspend their credulitysimilar to the way an audience in a classical theaterwas
supposedto suspendits disbelief arisingfrom unrealisticcostumes and surroundings. That is, a mathematicianmay believe a theorem assertedby a well-known
authorto be true,but may readthe argumentsin its proof as if the statementof the
theorem were in question. This predilectionfor separatingauthorsfrom proofs
appearsto be reflectedin the way proofs arewritten,for example,in the avoidance
of autobiographicalcommentin publishedmathematicalresearch.Indeeda general
tendency to impersonalize mathematicaltexts, especially research papers, has
been notedin analysesof their"naturallanguage"aspects(Burton& Morgan,2000).
By the early 1990s, readingand writing were beginning to be seen as different
aspects of one unified process. Similarly,validationandproof constructionmight
be seen as differing aspects of a single process. As such, they are probablybest
learnedin a dialecticalway. On the one hand, one constructsa proof with an eye
toward ultimately validating it and may often validate parts of it during the
constructionprocess. In fact, the final portionof a proof constructionis likely to
be a validationof that proof. On the other hand, proofs are writtenfor idealized
readers,yet there is considerablevariationin what actualmathematiciansknow.
Thus, the validationof a proof is likely to requirethe constructionof subproofs.
That is, each process, validationand proof construction,entails the other.
Validation and proof construction also differ in importantrespects. Proof
constructionis much more like mathematicalproblem solving, in the sense of
2 The idea that
meaningcan reside in the text was not new to the 20th century.It appearsto go back
at least to the time of Plato andbecame a majortenet for MartinLuther,who assertedthatthe meaning
of Scripturewas to be found in a deeperreadingof the text (Olson, 1977, p. 263).
3 In the context of this article,when mentioningmathematiciansor the mathematicscommunity,we
are referringespecially to those engaged in research in pure mathematics,that is, in finding and
proving theorems.
Annie Selden and John Selden
7
Schoenfeld(1985), thanis validation.Generallyconstructinga proof requiresthat
more diverse ideas come to mind at the "righttime"thanvalidatingit does. Also,
like most reading, validationnormallyemphasizes proceedinglinearly from the
beginningto the end of a writtenproof-perhaps repeatedseveraltimes.This linear
orderis unlikelyto occurin proofconstruction.Given a theoremto prove,one must
often attendnot only to the beginningbut also to the end of a proof before developing the middle. In addition,many proofs have a hierarchicalstructurebased on
subproofsandsubconstructionsthatemergesduringthe processof proofconstruction. Even assumingan idealized prover(one makingno mistakes or false starts,
which is an inherently unrealistic assumption), such an emergent hierarchical
structuremay lead to generatingsubargumentsin an ordervery differentfromthat
of the final writtentext. Indeed,for the workingmathematician,a theoremand its
proofsometimesemergein a dialecticalway over a long periodof time,with adjustments in one alternatingwith adjustmentsin the other.
One distinctionbetween currentideas of reading and that of validationis that
writtenproofs are still seen as carryingconsiderablemeaning. The proofs themselves (as opposed to theirreadersor authors)are regardedby mathematiciansas
the ultimateestablishersof the truthof theorems.4One examines neitherthe life
andtimesof a proof s authornorthe sophisticationof its readersin judgingthe truth
of a theorem. Perhapsthis is partly because any expanded or altered argument
constructedby a validatoris often seen as closely alignedwith the originalwritten
proof, so thatthe originalproofis deemedcorrect(in some generalsense) provided
the expanded,validator-generatedtext is seen by the validatoras correct.
Anotherdistinctionbetween validationandreadingin generalappearsto be the
unusualdegree of agreementaboutthe correctnessof argumentsand the truthof
theoremsarisingfrom the validationprocess.It is ourexperiencethatif two mathematiciansdisagree on the correctnessof a proof, they will often attempta joint
validationof it (or a fragmentof it). Typically, they will either expand the proof
and agree on the expandedversion or, failing that, they will find and agree on a
mistakethatcannotbe fixed. Occasionally,they may find andcorrectsmall errors
thatthey agreeareinsignificantanddo not affectthe overallcorrectnessof the argument.This abilityto reachagreementappearsto dependto some extenton the way
that mathematicaldefinitions have come to be treated as more or less unchallengeable in the 20th century.
Today, definitionsin mathematicsare typically treatedas analytic,as reducible
to undefinedtermsexcept for foundationalconsiderations,such as the meaningof
set, element, or integer.Such considerationsare not majorissues for most mathematiciansperhapspartlybecause "normal"foundationsyield a mathematicsthat
can be interpretedas parallelingan intuitiveview of the world developed socially
4 This view is supportedby the ability of computersto check the correctnessof many (more
formally stated)proofs. For example, a large numberof proofs have been machineverified after
conversionto the Mizarlanguageand are publishedonline in the Journal of Formalized
Mathematics(see mizar.org/JFM/).
8
Validations
as Texts
of ProofsConsidered
from an early age. Withboth "normal"foundationsand an intuitiveview one can
consider several objects together,count them, add, measurethings, and so on.' In
contrastwith analyticdefinitions,syntheticdefinitions6arethe everydaydefinitions
thatare commonly found in dictionaries-they areoften ill-specified descriptions
of thingsthatalreadyexist. It may be unclearwhen such everydaydefinitions(e.g.,
of democracy)are "complete"or whetherattentionto all aspectsof them is essential for their properuse. Analytic definitions, however, bring concepts into existence-the concept (e.g., of group as used in mathematics)is whateverthe definition says it is, nothing more and nothing less. One cannot safely ignore any
aspects of such definitions.There is a sense in which syntheticdefinitions can be
"wrong," that is, they can be inaccurate descriptions, but analytic definitions
cannot, since nothingis being described.'
For example, our syntheticdefinition of validation could be wrong; we might
have inaccurately described what we claimed is a recurringphenomenon. In
contrast, few mathematicianswould say the analytic definition of "group"was
wrong-uninteresting, tasteless,lackingapplications,or inaccuratelyremembered
perhaps,butnot wrong.Today'suse of analyticdefinitionsrendersvalidations(and
proofs andtheorems)very reliable,whereasthe earlieruse of syntheticdefinitions
sometimes left them problematic. This situation can be seen in Proofs and
Refutations(Lakatos, 1976) where validationsappearinherentlyunreliable,that
is, unreliableeven if there are no errors.There, theoremshave proofs and refutations (counterexamples)apparentlydueto treatingdefinitionsas syntheticandhence
challengeable. Given some literary license, Proofs and Refutationsprovides a
fairlyaccuratedescriptionof the way thatsome mathematicsdevelopedhistorically.
It illustrateshow definitionsandresultscan coevolve, but the compressionof time
involved in its fictional narration,togetherwith its synthetictreatmentof definitions, may suggest thatvalidations,proofs, and theoremsarefar less reliable than
they really are today.
Roles Played by Validation
Althoughwe arefocusing here on the validationabilitiesandpracticesof undergraduates who are in at least their 2nd year of mathematics study, validation
5 However, nothingin principlerestrictsmathematiciansto particularfoundations.For example, in
intuitionist/constructivistmathematics,one forgoes mathematicalinductionand the logical law of the
excluded middle, resultingin a more limited mathematicswithout proofs by inductionor contradiction.
6 Our distinction between analytic and
synthetic follows that made by Sierpinska, Defence,
Khatcherian,and Saldanha(1997) in discussing linear algebra.Similar distinctionshave been made
by Edwards(1997) using lexical andlogical andby Freudenthal(1973) using constructiveanddescriptive. In consideringdefining as a mathematicalactivity in geometry,Rasmussenand Zandieh(2000)
and de Villiers (1998) adoptedFreudenthal'sterminology.
7 Analytic definitionscan, however, be inspiredby regularitiesin the physical world, in which case
a mathematicalstructuremay be metaphoricallyrepresentedin a physical situation.Vectors areuseful
in navigationbecause the motion of a sailboat(in part)"workslike" vector addition.
AnnieSeldenandJohnSelden
9
appearsto have a role to play throughoutmathematicsstudents'education,in mathematicians'practice,and in examinationsof the natureof mathematics.
Validationin mathematicseducation.Holdersof bachelor's degrees in mathematics arenormallyexpected not only to know considerablemathematicscontent
but also to be able to constructmoderatelycomplex proofsandto solve moderately
nonroutineproblems. Indeed, one major way that an individual's mathematical
knowledge of a theorem is sometimes taken to be warrantedis by the ability to
"produce"a proof,not in a roteway butin the way a mathematicianwould produce
it and with understanding(Rodd, 2000). However, constructing or producing
proofsis inextricablylinkedto the abilityto validatethemreliably,and a proofthat
couldnotbe reliablyvalidatedwouldnotprovidemuchof a warrant.Solvingmoderately nonroutineproblems also appears to call for abilities akin to validation,
because one must ascertainwhethera proposedsolution is correct.In ourongoing
investigationof mathematicians'views, we arefindingthatmanymathematicians
believe thatchecking the correctnessof solutionsultimatelydependson the same
kind of deductivereasoningthatis used in proofs and their validations.
Preservicesecondarymathematicseducationmajorsand mathematicsteachers
also needto be ableto validateproofsreliablybecauseschool mathematicscurricula
are likely to place increasingemphasis on proofs and problem solving (NCTM,
2000). In this regard,Cuoco has observed informallybut on the basis of considerableexperiencethat"Thebest high school teachersarethosewho have a researchlike experiencein mathematics"(2001, p. 171).And in an articlein theMathematics
Teacher,Thompson(1996) suggeststeachingindirectproofto high school students
using number-theoreticstatementsthatare similarto, althoughperhapssomewhat
less complex than,the theoremused with undergraduatesin ourexploratorystudy.
Validationin thepractice of mathematicians.In additionto being importantfor
mathematicsmajorsandmathematicsteachers,validationappearsto play a fundamentalrole in the productionof new mathematics.A mathematician'sbelief in the
reliabilityand unproblematicnatureof validationprovides the assuranceneeded
to use a theoremin laterworkwithoutrepeatedlycheckingits correctness.Thatis,
when a theoremis proved "it stays proved."This reliabilityseems to have been a
majorcontributorto the rapidgrowththathas characterized20th-centurymathematics. Furthermore,many mathematiciansseem to rely mainly on validation,
sometimescarriedout individuallyand sometimesin seminars,to learnnew mathematics from the publishedresearchof others.
Validationwithinsocial constructivist(and other)perspectives. Ourcomments
aboutthe truthof theoremsmight suggest that we are taking a Platonistperspective, with its timeless abstractobjects and absolutetruthindependentfrom human
considerations.However, in this article, we are not favoring any single perspective on mathematics.Validationis a largelymentalprocess with social originsand
social consequences thatbears examining relative to any of several views on the
nature of mathematics. Furthermore,for many mathematicians,the truth of a
theorem appearsto be functionally just the existence of an argumentthat they
10
Validationsof Proofs Consideredas Texts
personally(or theirpresumedsurrogates)have validatedanddeemedto be a proof
of thattheorem.In particular,a furtherinvestigationof validationmightbe helpful
in viewing mathematicsfrom a social constructivistperspective-where mathematics is seen as an entirely social, fallible' phenomenonwith the idea of truth
replaced by social agreementand where abstractPlatonic objects have no role
(Ernest,1998).An accuratedescriptionof how new mathematicsis createdis crucial
from this perspective because, in a sense, how it is done is mathematics.We
suspect that, if analyzed together, proofs and validations thereof might in the
future play a larger role within a social constructivist perspective. In such an
analysis, foundationsandthe kind of logic used could be treatedas hidden,thatis,
tacitly assumed,premisesof theorems.By examiningthe extent of agreementand
the way it arises from the validationprocess, one might find a way to see mathematics as locally9infallibleand therebyreconcile this philosophicalview with the
more-or-less Platonic working views of most researchmathematicians.Indeed,
those workingviews may have emergedmainlyfromexperiencewith the reliability
of validation,ratherthanfrom any priorphilosophicalcommitment.
Regardless of one's perspectiveon the natureof mathematicsor its knowing,
investigations of validation as a form of reading (and of the argumentsbeing
read),with emphasison what validatorsdo, could informmathematicseducation.
Ourexploratorystudyis a startin this direction.We begin with an analysisof some
specific arguments.
A TEXTUAL ANALYSIS OF A THEOREMAND FOUR "PROOFS"
In this section, we analyze a theoremandfour undergraduatestudent-generated
argumentspurportedto prove it. This analysiswill provideconcretegroundingfor
the above theoreticaldiscussion of argument,proof, and validation,as well as for
the subsequentdescriptionof ourexploratorystudyof otherundergraduate
students'
validationsof these same four arguments.It may also expose aspectsof these texts
thatmight otherwisebe missed. For example, we point out thatcertainportionsof
the first argumentare extraneousand as such have no effect on its correctnessas
a proof. Nevertheless,errorsin these extraneousportionswere noted and thought
to be importantby some of our studentvalidators.
The theoremand the four argumentsare treatedhere primarilyas texts, that is,
as independentof specific authorsand validators.We will, however, commentin
generalon how aspectsof these texts may have come to be writtenor might affect
readers,occasionallyillustratingthis pointwith commentsfromourstudentvalidators. The theoremis true and one of the argumentsis a proof of it, whereasthree
8 Here,
following Ernest (1998), we mean by fallible something more than just the (generally
accepted)humantendencyto make mistakes.
9 By locally we mean the mathematicsvaries according to the foundationsand logic used. For
example, a mathematicalconstructivistwould not accept the law of the excluded middle and, consequently, obtain different theorems than a mathematicianwho did accept the law of the excluded
middle. We are not referringto geographicor social variations.
AnnieSeldenandJohnSelden
11
arenot;thatis, as proofsthey areincorrect.These characterizations-true,correct,
proof-are widely regardedby mathematiciansas globalfeaturesof texts alonethat
canbe determinedwithoutknowingwho the authorsor readersare.Mathematicians
say thatan argumentproves a theorem,not thatit proves it for Smith and possibly
not for Jones, who are readersof the argument.This view is held despite the fact
that the texts are the productsof humanactivities, that their physical representations could not exist without authors,and that validatorsare requiredfor anyone
to know the theoremis true.Perhapssuch assignmentsof proof, truth,andcorrectness (or their opposites) to texts is due to mathematicians'remarkablyuniform
agreementon whetherthese propertiesareassociatedwith particulartexts-at least
at the level of the mathematicsdiscussed here and modulo modest concernsabout
style or completeness.
Texts (and, more generally,language and communication)have been analyzed
fromseveralperspectives:syntacticstructure(e.g., Chomsky,1957), semanticstructure (e.g., Fillmore, 1968; Kintsch, 1974), and language in use (Halliday, 1977;
Vachek, 1966). Relevant ideas emphasizing language in use that are especially
applicableto mathematicaltexts have been summarizedandillustratedby Morgan
(1996). These ideas include such featuresas the ways reasoningis expressed(e.g.,
because, so), the symmetryor asymmetryof the relationshipbetween authorsand
readers,andwhattextsmightsuggestaboutthe natureof mathematics.Mostof these
features,for example the use of "we" in mathematicaltexts (Pimm, 1984), have
no bearingon whetheran argumentprovesa theorem;andin ourexperience,undergraduatestudents realize this and concentrateon the underlying mathematical
concepts and inferences.
Thus, we provide a differentkind of textual analysis, one that is a kind of lineby-line gloss, or elaboration,of the theoremand the four student-generatedarguments emphasizing mathematicaland logical points that a validator might, or
might not, notice. For example, in consideringthe statementof the theorem,we
discuss both its logical structureand other, more informal,ways that it could be
written.For the four student-generatedarguments,we discuss such mattersas the
use of alternativeterms(e.g., assume for suppose, divides insteadof multiple),the
role of individualsentences in furtheringthe argument,properand improperuses
of symbols, implicit assumptions(e.g., that a division of the argumentinto cases
is exhaustive), the correctness of inferences, computationalerrors, extraneous
statements, and structuralaspects of the argument.Global properties, such as
whetheran argumentproves the theorem,as opposed to some othertheorem,are
also discussed.
In the following analyses, the names of the four arguments(e.g., "Proof(a)":
Errors Galore) and the numberingof their parts are for the convenience of the
reader;the names and numberingwere not shown to the eight studentvalidators
in our study. The theoremto be proved was this:
THEOREM:
For any positive integer n, if n2 is a multipleof 3, then n is a multiple
of 3.
12
as Texts
Validations
of ProofsConsidered
Both the studentauthorsof the four argumentsandthe studentvalidatorsin our
empirical study were given this statementof the theorem.It is a formalstatement
in that the variable n, the set to which it refers (the positive integers), and the
universalquantification,any, areexplicitly stated,andthe conditionalis expressed
in its most common form, if. ..., then. However, other, less formal statementsof
this theoremconvey the same logical information;theirequivalencecan be established using logic andlittle or no additionalmathematicalcontent.Often,theorems
that undergraduatestudentsat this level consider are stated in a less formal way.
For example, one informal statementof this theorem is A positive integer is a
multipleof 3 wheneverits square is.
Undergraduatestudentssimilarto the ones in our empiricalstudyappearto have
considerabledifficultyinterpretinginformalstatementsof theoremsandconverting
themto formalones (Selden& Selden, 1995). In ourexperience,formalstatements
appearto be the ones most easily interpretedin validations;they correspondto what
we have called proof frameworks.These consist of the beginning and end of a
proof-the partthatcanbe writtenwithoutknowingthe meaningsof all of the mathematical terms and from which the theoremcan be reconstructed.It is an understanding(usuallytacit)of the relationshipbetweenstatementsof theoremsandtheir
proofframeworksthatallows validatorsto decideif an argumentprovesthe theorem
at handas opposedto some othertheorem.A proof framework'"
for this theoremis
this: PROOF.
Let n be a positive integerand supposen2 is a multipleof 3. ... Then
n is a multipleof 3. Often the introductionof a variable,for example, "Letn be a
positive integer,"is omittedbutunderstood.This omissionusuallyoccurswhen the
variableappearsin the statementof the theorem,as it does here. Ourearlierwork
(Selden & Selden, 1995) suggests that,amongthe variouspossible ways to statea
theorem, studentvalidatorsshould find the more formal statementused here the
easiest to correctlyconnect with argumentspurportingto prove it.
In what follows, we state each student-generatedargumentverbatimand then
give our line-by-line analysis of it.
"Proof (a)": Errors Galore
PROOF.
Assume thatn2 is an odd positive integerthat is divisible by 3. That is
n2 = (3n +1)2 = 9n2+6n +1 = 3n(n + 2) + 1. Therefore,n2 is divisible by 3. Assume
thatn2 is even and a multipleof 3. That is, n2 = (3n)2 = 9n2 = 3n(3n). Therefore,
n2 is a multipleof 3. If we factorn2 = 9n2, we get 3n(3n); which means thatn is a
multiple of 3. U
Analysis of "Proof (a)"
[1] Assume that n2 is an odd positive integer that is divisible by 3. This partof
the argumentis not wrong. In this context one can take assume to mean suppose,
10Proof frameworksare often more
complex thanthis. For examples from calculus and semigroup
theory, see Selden and Selden (1995, pp. 129-130).
AnnieSeldenandJohnSelden
13
and an authorof a proof can suppose any reasonablenon-selfcontradictorystatement which might aid the progressof the argument.But, as some of our student
validatorsnoted, this division into odd and even cases is a peculiarway to structure a proof of this particulartheorem.It is peculiarbecause odd and even appear
to be unrelatedto multiplesof 3.
That the phrase "n2 is divisible by 3" is an equivalent way of saying "n2 is a
multipleof 3" is somethinga validatorwould need to assent to (by noting thatthe
definitionsof divisibilityandmultiplearemathematicallyequivalent).One student
validator,whom we refer to in this articleas SW, briefly noted (and assentedto)
the fact that "divisible by 3" was used here; subsequently, in the even case,
"multipleof 3" was used.
[2] Thatis n2 = (3n + 1)2 = 9n2 + 6n + 1 = 3n(n + 2) + 1. There are four errors
here.First,the assumptionof oddnesswas aboutn2, not n, to which it is apparently
being applied (i.e., n = 3n + 1). Second, n is being used in two places with two
differentmeaningswithinone equation;in effect, the authoris mistakenlyasserting
thatn = 3n + 1. In the genre of mathematicalproofs, it is not permissibleto let the
samesymbolrepresenttwo differentnumbersexceptacrossindependentsubproofs.
Doing so would be very likely to cause validatorsconfusion. Indeed, studentST
remarkedthat "all those n's tryingto equal each other"misled her. Third,an odd
integern can be representedas n = 2m + 1, not as 3m + 1, where m is a different
integerfromn. Fourth,the last equationis incorrect;3n times n yields 3n2,not 9n2.
We would like to commentfurtheron the author'sapparentidea that3n + 1 can
representa generalodd number.We doubt that this idea indicatesa more-or-less
fixed misconceptionin the author'sknowledgebase,butrathersuspectit is an incorrect,partialrecollectionof the idea that2n + 1 is odd for integervalues of n. There
appearsto be a more subtle lack in the author'sknowledge, and we will offer a
conjectureof its natureandhow it occurs.Studentsof mathematicsprobablyrecognize (mostlytacitly)manykindsof problemsandtasks,andeach kindmay be associatedwith manyotherideas, forminga mentalstructurelike a conceptimage(Tall
& Vinner, 1981) for thatproblemor task-what we have called a problem-situation image. When a kind of problemor task is recognized,an associatedproblemsituationimage may be activated(broughtto mind) or partlyactivated(Baddeley,
1995). In particular,accomplishedstudentsmightassociatethe taskof representing
a generaloddnumberwith a feeling of cautionaboutgettingtherepresentation
right.
As they execute the task,this feeling of cautionmay engenderchecking.In the case
at hand,thatmight mean substitutingsome numbersfor n: substituting1 for n (in
3n + 1) yields4 andindicatesthat3n + 1 neednotbe odd.We suspectthatthe author's
problem-situationimage(s) for this task did not include an appropriatefeeling of
caution.A mechanismfor bringingsomethingto mind similarto the one sketched
here, concerninghow to startsolving a recognizedkindof problem,has been more
fully describedin Selden, Selden, Hauk,& Mason (2000).
[3] Therefore,n2 is divisible by 3. It was assumedin [1] thatn2 is divisible by 3.
Thus,this statementcannotbe wrongdespitethe interveningerrorsin [2]; however,
it cannotbe of use either.Furthermore,this statementcompletes one of two inde-
14
Validations
as Texts
of ProofsConsidered
pendentsubproofs,one for n2 odd andone for n2 even, and shouldhave endedwith
n as a multiple of 3, not n2.
The errorsin [2] might be called extraneouserrors.Because they do not affect
the correctnessof [3], they cannot affect whetheror not the argumentis a proof.
Such errors are nevertheless undesirable because they can make arguments
confusing and more difficult to validate. In our empirical study, most student
validatorsspent considerabletime trying to decipher [2], making comments like
"3n + 1 doesn't make sense [for odd],"but did not seem to notice its irrelevance.
[4] Assumethatn2 is even and a multipleof 3. This assumptionis an appropriate
beginning of the second independentsubproof(case). However, as alreadynoted
under [1], this division into cases, althoughnot wrong, is unlikely to be useful.
[5] Thatis n2 = (3n)2 = 9n2 = 3n(3n). In the first equationn2 = (3n)2, it appears
thatthe assumptionthatn2 is a multiple of 3 is incorrectlyappliedto n insteadof
n2. Also, in the implicitequationn = 3n, once againthe right-handn does not have
the same meaningas the left-handn. To indicatethatn is a multipleof 3, one represents n as 3m, wherem is anotherpositive integer.The second and thirdequations
are correct.
[6] Therefore,n2is a multipleof 3. This conclusionis not incorrect;it does follow
from [5] (if one replaces all but the first "n"with "m's"). But it also just repeats
the assumptionin [4], which is not likely to be useful.
[7] Ifwefactor n2 = 9n2,we get 3n(3n).Here the authorappearsto be attempting
to restateinformationfrom [5], which could not introduceany new errors.Since
the word "factor"refersto expressions,not equations,we take it that[7] is a shortened form of "Now n2 = 9n2, and if we factor9n2, we get 3n(3n)."
[8] whichmeans thatn is a multipleof 3. U This inferencefrom [7] is not correct
even if we change3n(3n)to 3m(3m).To concludethatn is a multipleof 3, one would
expect to see 3 as a factorof n, not as a factorof n2.The symbol "N" is one of those
customarilyused to indicatethe end of a proof.
In summary,this argumentis not a proof. It consists of two independentsubargumentseach of which should end with "n is a multipleof 3" or its equivalent,
"nis divisible by 3." However,the odd case did not end this way, andthe even case
made this claim but did not properlyjustify it. It is left implicit thatthe two cases
are exhaustive, thatis, thatevery integeris eithereven or odd, but it is customary
to leave such a well-known fact implicit.
We would like to mention two kinds of global errorsthat this argumentdoes
not contain. First, although the argumenthas a high density of errors,they are
not what one might call mathematically syntactical errors (Selden & Selden,
1987). The mathematicaltermsand symbols fit togetherin a meaningfulway (i.e.,
one canjudge whethereach individualstatementis corrector incorrect).Second,
the two subargumentsreally are independent. Occasionally, one finds arguments in which an inference in one subargumentimproperlydepends on information from inside anothersupposedly independentsubargument,and thaterror
did not occur here.
AnnieSeldenandJohnSelden
15
Last, we have taken this argumentat face value; that is, we have assumed that
the authormeantto write a proof andhad some more-or-lessclear (althoughincorrect) conception of how the argument would proceed. However, because the
studentauthorsubmittedthis "proof' as partof a take-hometest, it is possible that
he or she was merely seeking some (partial)credit and was thus workingin what
Vinner(1997) has called a pseudoconceptualmode," mimickingsome previously
encounteredargumentaboutmultiplesof 2. One possible similarargumentmight
have been the following proof of the statement:For any integern, n2 is a multiple
of 2 if and only if n is a multipleof 2.
PROOF.If n is odd, then n2 = (2k + 1)2 = 4k2 + 4k+1 = 2(2k2 + 2k) + 1, which
is odd. Therefore, if n2 is a multiple of 2, then n is a multiple of 2. Now, if n is
a multiple of 2, that is, if n is even, then n2 = (2k)2 = 4k2 = 2(2k2), which is a
multiple of 2. M
"Proof(b)": TheReal Thing
PROOF.
Suppose to the contrarythatn is not a multipleof 3. We will let 3k be a
positive integer that is a multiple of 3, so that 3k + 1 and 3k + 2 are integersthat
are not multiples of 3. Now n2 = (3k + 1)2 = 9k2 + 6k + 1 = 3(3k2 + 2k) + 1. Since
3(3k2+ 2k) is a multipleof 3, 3(3k2+ 2k) + 1 is not.Now we will do the otherpossibility, 3k + 2. So, n2 = (3k + 2)2 = 9k2 + 12k + 4 = 3(3k2 + 4k + 1) + 1 is not a
multipleof 3. Because n2 is not a multipleof 3, we have a contradiction.E
Analysis of "Proof(b)"
[1] Suppose to the contrary that n is not a multiple of 3. The phrase "to the
contrary"may indicatethatthis argumentis meantto be a proof by contradiction.
It mightalso indicatethatthe argumentwill provethe contrapositiveof the theorem,
thatis, thatthe negationof the conclusion implies the negation of the hypothesis,
which is logically equivalentto the theorem.The studentauthormay not be clear
aboutthe distinction.If the proofis reallyby contradiction,it is implicitthatn represents a positive integerand n2 is assumedto be a multipleof 3. The argumentcan
end with any contradiction,but thatcontradictioncould be to the hypothesis,that
is, n2 is not a multipleof 3. If the contrapositiveof the theoremis being proved, it
is implicitonly thatn representsa positive integer,andthe argumentmustend with
n2 is not a multipleof 3. Forboth methodsof proof it is appropriateto supposethe
negation of the conclusion, that is, n is not a multipleof 3, as is done here.
[2] Wewill let 3k be a positive integer that is a multipleof 3, so that 3k + 1 and
3k + 2 are integersthatare not multiplesof 3. This representationanddivision into
cases correctlyfollows from the fact thatany positive integercan be represented
as exactly one of 3k, 3k + 1, or 3k + 2 for some positive integerk. It is implicit that
11 Indeed, one student validator,ST, seems to have concluded that the authorof "Proof (a)" was
workingpseudoconceptuallyor at least had no clear conceptionof a proof. After finding severalerrors
and noticing thatcertainpartsdid not make sense, she stated,"Basically,it's a bunch of nothing."
16
Validations
as Texts
of ProofsConsidered
the positive integerbeing representedis n andthat,accordingto [1], n cannotequal
3k; thus, n must equal 3k + 1 or 3k + 2. A validatorwould need to check that this
statement is correct. To comprehend [2], it would not be essential to see the
implicitrepresentationof n as one of the threeexpressions,but it would be in order
to comprehend[3].
[3] Now n2 = (3k + 1)2= 9k2 + 6k + 1 = 3(3k2 + 2k) + 1. This statementbegins
the first of two independent,"parallel"subarguments.Assuming that n = 3k + 1,
all of the equationsare correct.
[4] Since 3(3k2 + 2k) is a multipleof 3, 3(3k2 + 2k) + 1 is not. It follows from
considerationssimilarto those under[2] thatonly every thirdpositive integeris a
multiple of 3. So if an integer is a multiple of 3, then one more than it is not. A
validatorwould need to assent to this. It is then implicit in [4] that3(3k2+ 2k) + 1
is the n2 mentionedin [3]. A validatorwould not have to notice this to comprehend [4], but would need to see it to comprehend[5].
By a validatorassentingto something,in this case that 3(3k2 + 2k) + 1 is not a
multiple of 3, we mean that the validatoris conscious of agreeing with it. This
conscious agreementmay be fleetingandnot focusedon, butit is distinctfrommere
passive readingin inner speech. It may be the only conscious phenomenonassociated with a fairly complex inference.In this case, the validatormust have some
knowledge of the relationshipbetweenmultiplesandmultiplesplus one to see that
3(3k2+ 2k) + 1 fits thatrelationshipandinfer that3(3k2+ 2k) + 1 is not a multiple
of 3. While the assentingis conscious, the inferenceleading to it may occur very
quickly and outsideof consciousness. It would be a validationerrorfor a validator
to passively read, ratherthan assent to or reject, a portionof a proof, even if the
proof were correct and the validatoragreed that it was correct. Such a validator
would not have tested thatportionof the proof and could not really know that the
proof was correct.
[5] Now we will do the otherpossibility, 3k + 2. This assertionindicatesthatthe
first subargumentis finished and that the second subargumentis beginning with
the correct,but implicit, assumptionthatn = 3k + 2.
[6] So, n2 = (3k + 2)2 = 9k2 + 12k + 4 = 3(3k2 + 4k + 1) + 1 is not a multipleof
3. All of the equationsare correct.The final assertiondependson the same informationused to justify [4] and should be assentedto by a validator.This statement
concludes the second subargument.
[7] Because n2 is not a multipleof 3, we have a contradiction.MThis statement
is correct, assuming that it is a proof by contradiction.The implicit reasoningis
that, according to [2], one of the two subarguments(cases) must apply to n2.
Because both subargumentsyield that n2 is not a multiple of 3, there is a contradiction to the implicit assumptionthatn2 is a multipleof 3. If this argumentwere
treated as a proof of the contrapositive, it would be peculiar to mention "the
this final stepcould be omitted
contrary"in [1]. However,underthatinterpretation,
entirelyor replacedby "Thus,in eithercase, n2 is not a multipleof 3." We regard
this argumentas a proof of the theorem,althoughit mightbe writtenmore clearly.
If the role of n2 and the division into two independentsubarguments(cases) were
AnnieSeldenandJohnSelden
17
made explicit ratherthanimplicit in [2], [3], [4], [5], and [6], this proof would be
less confusing for a validator,especially an inexperiencedone.
"Proof(c)": The Gap
PROOF.
Let n be an integer such thatn2 = 3x wherex is an integer.Then 31n2
Since n2 = 3x, nn = 3x. Thus, 31n.Thereforeif n2 is a multiple of 3, then n is a
multipleof 3. E
Analysis of "Proof(c)"
[1] Let n be an integer such that n2 = 3x where x is an integer. Except for the
omission of the word "positive,"this representationis an appropriateway to begin
a proof of this theorem.However, it is very abbreviated,and for mathematicsat
this level, the argumentwould be more clearlylinkedto the theoremby supposing
thatn2 is a multipleof 3 before claiming n2 = 3x, wherex is an integer.
[2] Then31n2. This inferenceis correct,but not likely to be useful. It is read"3
divides n2"and is just anotherway of saying n2 = 3x, wherex is an integer,which
was the suppositionin [1].
[3] Since n2= 3x, nn = 3x. The inferenceis not wrong,but it is stilljust the informationof [1] writtenin a slightly differentform.
[4] Thus,31n.Read "3 divides n," this statementmeans thatn is a multipleof 3.
Nothing in [1], [2], or [3] indicateswhy [4] should be true.Indeed,to fill the gap
would amountto proving the theorem.
[5] Thereforeifn2 is a multipleof 3, thenn is a multipleof 3. U This conclusion
is just a restatementof the theorem. The argumentreally ends with [4]. If the
reasoningfrom [1] to [4] were correct(which it is not), [5] would be true. In the
genre of mathematicalproofs at and beyond the collegiate level, a theoremis not
normallyrestatedat the end of an argumentpurportingto prove it. However, this
sort of summarysentence can be found in sampleproofs in transitioncourse textbooks, especially when an argument for the contrapositive has been given
(Velleman, 1994).
"Proof(d)": The Converse
PROOF.
Let n be a positive integer such that n2 is a multiple of 3. Then n = 3m
where m e Z+. So n2= (3m)2= 9m2= 3(3m2).This breaksdown into 3m times 3m
which shows thatm is a multiple of 3. 0
Analysis of "Proof(d)"
[ 1] Let n be a positive integersuch thatn2 is a multipleof 3. This representation
andsuppositionis an appropriateway to begin provingthe theorem.Beginningthis
way, the theoremwould be proved if the argumentled to n is a multipleof 3.
[2] Thenn = 3m where m e Z+. Here Z+ representsthe set of positive integers.
Statement[2] is incorrect.The meaningof multipleand [1] yields n2 = 3m, where
18
Validations
as Texts
of ProofsConsidered
m e Z+, but [2] does not follow directlyfrom [1] (except by an applicationof the
theorembeing proved).
[3] So n2 = (3m)2= 9m2= 3(3m2).All threeequationsare correct,given [2].
[4] This breaksdown into 3m times 3m This statementfollows from [3].
[5] which shows thatm is a multipleof 3.E This inferencedoes not follow from
[4] or anythingelse in the argument.The authorprobably"meant"thatn2, rather
thanm, is a multipleof 3. When changed, [3] yields thatn2, not n, is a multipleof
3, which means the conversehas been proved(by [2], [3], [4], andthe altered[5]).
All four of the undergraduatestudent-generatedargumentsabove correctly
use a logical andrhetoricaldevice thatis particularlycommonin the genreof mathematical proofs. That is, to prove a theorem claiming some propertyabout all
(each) of the elements in some set, for example, about all positive integers, it is
sufficient to argue about a particular,but arbitrary,element of the set. In the
language of secondary school algebra, one might say that the argumentis made
about an unknown constant rather than a variable. Typically, the element is
namedvery earlyin the argumentto avoid the possibilityof inadvertentlycompromising its arbitrariness.At the conclusion of the argument, it is implicit that
whichever element of the set might be of interest,the argumentcould have been
about that element. Although this rhetoricaldevice is simple, it is very powerful
and simplifies the logic requiredof validators, thereby perhaps avoiding some
errors.In the historical development of calculus, this device together with carefully formulated,analytic definitions (like the epsilon-delta definition of limit)
allowed theoremsabout motion (for centuries a difficult concept) and, in particular, about rate of change, to have proofs in which nothing changes and motion
is not mentioned.
The textual analysis above differs from a descriptionof actual validations. It
providesbackgroundinformationaboutthe theoremand the four "proofs"useful
for understandingvalidationsof them. However, it does not include descriptions
of actual validators' mental processes, abilities, missteps, and so on, in their
attemptsto determinewhich of the argumentsarereally proofs. We now describe
how some undergraduatemathematicsand mathematicseducationmajorsjudged
whetherthese same four argumentswere proofs.
THE EXPLORATORYSTUDY OF VALIDATION
Eight undergraduates(four secondaryeducationmathematicsmajorsand four
mathematicsmajors)at a comprehensivestateuniversityin the United Stateswere
interviewed individually outside of class in the 4th and 5th weeks of a 15-week
"bridge" or "transition to proofs" course. Although this course is normally
studied in the 2nd or 3rd year at university, three of the eight students were in
their4th year. The eight formed the entire cohort of this learning-to-prove-theorems course requiredof all mathematicsand secondary mathematicseducation
majors at the institution.The goal of such courses, offered by many U.S. mathematics departments,is to help students make the transition from an earlier
AnnieSeldenandJohnSelden
19
traditionalcalculus sequence to later, more proof-basedcourses, such as abstract
algebra and real analysis. They typically cover various aspects of logic; a little
aboutsets, relations,andfunctions;proofby mathematicalinduction;andperhaps
a few additionaltopics, such as elementarynumbertheory or introductorygraph
theory, about which students are asked to prove theorems. The emphasis is on
having studentslearnto make theirown proofs as opposedto the amountof mathematics covered. Kurtz'sFoundationsofAbstract Mathematics(1992), a typical
textbook, was used in the course. In addition, students were to spend much of
their class time, in pairs, proving theorems at the chalkboard.All eight students
had taken single-variable calculus; some had also taken multivariablecalculus,
differentialequations,matrixalgebra,or discretestructures.One student(referred
to as BH) was concurrently enrolled in a dual-listed graduate/undergraduate
numbertheory course.
TheStructureof the Interviews
Each studentwas interviewedindividuallyfor about 1 hourin a semistructured
mannerusing an interview scriptconsisting of four phases.
Phase 1. During this phase, which could be termed the warm-upphase, each
studentwas given a brief fact sheet (see Appendix 1) and this writtenstatement:
Forany positive integern, if n2 is divisible by 3, thenn is divisible by 3. They were
asked to explain in their own words what the statementsaid, to give some examples of it, and to decide whetherit was true and how they would know. Finally,
they were asked to give a proof of the statementif they could. Two successfully
did so. After some time had elapsed, those who could not complete a proof were
advisedthatthey need not continue,becauseprovingthe theoremwas not the point
of the interview; rather,they were to judge the correctness of other students'
"proofs"of the statement.Studentswere told thatsince some of the "proofs"they
would see might seem a bit unusual,having attempteda proof would help them
appreciatethe variety of approachestaken.
Phase 2. Eachstudentwas shownthe four"proofs,"one afterthe other,andasked
to think out loud as they read each one and decided whetherit was, or was not, a
proof. If it was not a proof, they were to point out which part(s)were problematic.
If theycould, they were to say where,or in whatways, the purportedproofhadgone
wrong.They were also informedthatthese "proofs"had been submittedas partof
a take-hometest in a previousyear by transition-coursestudentslike themselves.
The four "proofs"were presentedto the studentsas they appearin Appendix2.
They were told to take as long as they liked and seemed not to be hurried.For
example,one student(JB) spent 15 minutesreadingandrereadingthe first"proof'
before deciding (incorrectly)that it was indeed a proof, except for a minornotational error.Several of the studentsdid not want to commit themselves and were
allowed duringthis phase to say thatthey were unsurewhethera given argument
was, or was not, a proof.
20
Validations
as Texts
of ProofsConsidered
Phase 3. Having seen andthoughtaboutall four "proofs"one afterthe other,the
studentswere thengiven an opportunityto rereadthemall togetherandrethinktheir
earlierdecisions with an opportunityto change theirminds, and some did. At the
end of this phase,they werenot allowedto say thatthey wereunsure,butwere asked
to make a decision and declarewhethereach argumentwas, or was not, a proof.
Phase 4. Finally, the studentswere asked eight generalquestions aboutproofs
and how they read,understand,and validateproofs.
The data collected consisted of the audiotapedinterviews, the interviewer's
notes, andthe students'workon the "proof' sheetsprovidedduringthe interview.
(See Appendix 2 for details.)
The Conductof the Interviews
The interviews were treatedas "teachinginterviews."Not only was the interviewer the instructorfor the transitioncourse thatthe studentswere taking,it also
seemed appropriateand opportuneto the interviewer-instructorto answer their
implicit andexplicit questions.The interviewerwould respondgenerallyaboutthe
structureor style of writtenproofsbutneverindicatewhetherthe fourspecific arguments thatthe studentswere examining were actuallyproofs of the theorem.This
procedurekeptthe tone of the interviewsas relaxedandconversationalas possible.
In addition, as the interviews proceeded, the students sometimes seemed to
need encouragementto keep readingand thinkingaboutthe four "proofs."Thus,
to obtain as much informationas possible, the interviewerintermittentlymade
comments and answered or asked questions to keep students on task. All the
studentsseemed to take the interviews seriously and seemed genuinely engaged
with the task at hand.
Informingstudentsof mathematicalpractices and norms.Keeping the students
on task occasionally had the effect of suggesting to them practices and norms of
the wider mathematicscommunity.In the following excerpt,studentMM hasjust
begun to consider"Proof(a)"for the first time. She examines it silently for 1 minute, then asks whetheranother3 should be in the expression 3n(n + 2) + 1. With
encouragementfromthe interviewer,she makesher suggestedcorrection,writing
9n2 + 6n + 1 = 3n(3n + 2) + 1. She then continues reading silently for an additional 30 seconds. Then, the following interchangeoccurs.
MM. I don'tlikethisone 'causeit's confusing[laughs].
L
OK,youdon'tlikeit.Inadditionto beingconfusingandnotlikingit-whichis a very
goodreasonnotto likeit andwhichis whyI'mgoingto insistyouall [meaningthe
studentsin thetransition
class]writenonconfusing
proofs[bothlaugh].OK?
MM. There'sanerror.
I:
Ahh,there'sanerrorthere..... Let'spretendthe studentmeantto writethat[indi9n2+ 6n + 1 = 3n(3n+ 2) + 1]....
catingMM'scorrection
MM: OK.
I:
... andsee if thatmakesa proof.
AnnieSeldenandJohnSelden
21
The norm12
being suggestedhere is that stylistic clarity(i.e., writingproofs in a
nonconfusingway) is valued,butthatlack of clarityin itself does not makean argument incorrect.In addition,it is impliedthroughthe overtact of correctingan algebraic errorthat small errorsin writtenproofs (e.g., those similarto misprints)can
be fixed duringthe validationprocess.
In the following example, studentST is in the midst of considering"Proof(a)."
ST: Theyalsohaven2= (3n)2.Theyuse the samevariableforeverythingso it kindof
throws(?) everythingtogether.
[STandtheinterviewer
agreeto fix thisby writingn2= (3m)2.]
ST: I thinkthat'swhat,... that'swhat,... threwmeoff atthebeginningbecauseI was
lookingat all thosen's tryingto equaleachother.
intherethatyoudon't,uh,usethesame
I:
That'sreallyhard,isn'tit?There'sa principle
letterto meantwo differentthingsin one proof.Youcanfinisha proof-and then
the idea is finished-and thenuse the letterto meansomethingnew, but that's
becausethere'sa breakin theideas.
In this excerpt, ST comments on the confusion caused by using n in several
differentways. The interviewercorroboratesthisandgives an additionalbit of informationregardingwhen it is permissibleto use the sameletter,suchas n, again.This,
andothersimilarcommentsmadeduringthe interviews,hadtheeffect of introducing
these studentsto some stylistic featuresof writtenproofs at the same time as the
studentssaw (for themselves) a rationalefor those bits of mathematicalpractice.
Furtherreflectionencouragedby the interviewer'sremarks.All studentswere
kept on task by the interviewer'sdeliberateattemptsto probe their thinking.We
illustratethis procedurewith comments from KC, a particularlyexpressive and
responsivestudent.She told the interviewerthatshe ordinarilylikes to readproofs
aloud to herself, that is, she had no problem converting inner speech to regular
speech duringthe interview.Below is a condensed version of her first attemptat
validating "Proof (d)." It proceeded along a tortuous sequence of meandering
stages, numbered 1-8 below, apparentlyassisted by the interviewer's probing
questions. (For the entiretranscript,see Appendix 3.)
1. KC readsthe entireproof aloud slowly once and remarksthat"Proof(d)"looks
a lot like "Proof(a)" and says, "I thinkthat's a proof."
2. On being questioned by the interviewer who is genuinely surprisedby this
analogy, she explains somethingabout 3(3m2) and tries to rewritethatportionof
the proof, concentratingon the form she wants to see, "threetimes some integer."
3. After some time, the interviewerasks whetherthat is good enough for a proof
12This kind of norm (of the wider mathematical
community)is reminiscentof sociomathematical
norms,the negotiationof which has been investigatedin classroomsettings,for example, at the undergraduatelevel by Yackel, Rasmussen,andKing (2000). It would be interestingto investigatehow such
normsaremaintainedand,in particular,adoptedby novices in the widermathematicalcommunity.The
kinds of discussions associatedwith these interviewsmay not occuroften enough in universitycourses
to explain the maintenanceof such norms.
22
Validations
as Texts
of ProofsConsidered
now, andKC commentsthather wordingmay not be right.She andthe interviewer
discuss this at some length and agree there are lots of equivalent ways to word
things.
4. The interviewerthen asks whethershe has finished, and KC respondswith an
explanation,". .. Let's see, they showed thatm was a multipleof 3 but they didn't
show n was a multipleof 3."
5. She triesto substitutebackin andthen suddenlyblurtsout, "OK,wait, wait ...,"
followed by 30 secondsof intensesilentconcentration.She thenthumpsherpencil
twice emphaticallyon the desk and asserts,"I don't thinkthat's a proof."
6. The interviewerasks why she changedher mind, and KC says, '"Theyshowed
that m was a multipleof 3, but they didn't show n was a multipleof 3."
7. The interviewerthen asks, "Wheredid that m come from anyhow?"to which
KC responds, ". .. they let n = 3m right up here.. ." [looking back to the beginning of the argument],and notes that they did not startin the right place, that is,
n2 = 3m.
8. She reiterates,"I don't thinkthat's a proof."
In respondingto the interviewerat stages (2), (3), (4), (6), and(7), KC is encouraged to reflectfurtheron "Proof(d)"andnotices featuresof the argumentthatwent
undetected earlier. However, it is not easy to conjecturewhat happenedto KC
during the 30 seconds of intense silent concentrationat stage (5). Her insight
cannot reasonablybe attributedto anythingspecific that the interviewerdid; she
appearsto have had somethinglike a genuine "Aha!"experience (Barnes,2000).
It would be very interestingto know whatKC was consciousof duringthis experience. We suspect such an experience might be described immediatelyafter its
occurrence,but is often not well rememberedlater.The interviewerdid question
KC immediatelyaboutwhy she changed her mind, but not aboutthe contents of
her conscious experience. To have done so might have considerablyalteredthe
overall validationprocess.
Overall Results
On analyzing the data, it became clear that most students made a judgment
regardingthecorrectnessof each"proof"atfourdifferenttimes,namely,atthebeginning and at the end of each of two readings(in Phases 2 and 3). These times were
labeled Time 1 throughTime 4. All students,except JB, maintainedor increased
the numberof theircorrectjudgmentsfromTime 1 to Time 4. (See Figure 1, which
displays the changingnumberof correctjudgmentsby each studentacross time.)
For example, KC graduallyimprovedfrom one correctjudgmentat Time 1 to four
correctjudgmentsat Time 4. This improvementappearsto be the resultof further
reflection.
Eight studentswere interviewedregardingthe four "proofs,"making a total of
32 person-proofjudgments.Table 1 gives thepercentageof correctjudgmentsmade
at the varioustimes.
Annie Selden and John Selden
4
3
2
1
0
SD
'
ST
23
JB
BH
I
KC
1
RC
'
MM
'
SW
Figure1. Thenumberof eachstudent'scorrectjudgmentsacrosstime(Time1 to Time4)
Table1
Percentage and Numberof JudgmentsOver Time
Correct
Incorrect
Unsure
Time1
46%(15)
25%(8)
28%(9)
Time2
56%(18)
28%(9)
16%(5)
Time3
72%(23)
22%(7)
6%(2)
Time4
81%(26)
18%(6)
0%(0)
Note.Thetotalnumber
ofjudgments
was32.
What happened over time that might have caused students to change their
minds?Duringthe second phaseof the interview,betweenTime 1 andTime 2, the
interviewerasked and answeredquestions,encouragingthe studentvalidatorsto
reflect further.By the beginningof the thirdphase of the interview,they had seen
and ponderedall four "proofs"and were more experienced.At Time 4, the interviewer would no longer accept "unsure,"and the studentsmade their final judgment for each "proof."
At Time 1, less than half (46%) of the students'judgmentswere correct.This
remarkablefact suggests that given this task on a test, where no one would have
been encouragingthem to reflect further,the studentswould probablyhave done
about as well by chance.3 They initially made the most correct judgments on
"Proof(b)," with five correctlystatingthatit was a proof and threebeing unsure.
They also initially made the fewest correctjudgments on "Proof(d)," with only
two makingthe correctjudgmentand four incorrectlystatingit was a proof. This
difficulty with "Proof(d)" supportsour observationbelow thatthe studentswere
difficulties.
primarilycheckinglocal detailsinsteadof looking for global/structural
After the first sentence, "Proof (d)" can be regarded as an argument for the
converse, modulo one notationalerror.The studentsalso seemed to be relying on
theirfeelings of understandingin a way thatsometimesmisled them. StudentMM
13 Althoughthe students'
ability to determinewhetherthese argumentswere proofs appearedto be
very limited,we arenot claimingthis is unchangeable.Indeed,we saw some evidenceof improvedjudgments even duringthe course of the interviews and conjecturethatthe students'initial lack of ability
was largely due to a lack of appropriateexperiences.
24
Validations
as Texts
of ProofsConsidered
commentedabout "Proof(d)" that she "liked it" and could "see it betterthan the
rest."Table 2 shows how the aggregatedstudents'judgmentson the fourpurported
proofs changedover time. The entriesrepresentnumbersof students.
Table2
Changes in the Patternof Judgments
"Proof(a)" "Proof(b)" "Proof(c)" "Proof(d)"
Correct
Incorrect
Unsure
4355
3433
1100
5678
0100
3110
4466
1112
3310
2557
4331
2000
Note. Entriesgive the numberof studentsresponding"correct,""incorrect,"or "unsure"at each of the
four times. For example, under"Proof(a)" for Correct,therewere 4, 3, 5, and 5 correctjudgmentsat
Time 1, Time 2, Time 3, and Time 4, respectively.
A cleartrendcan be seen for "Proof(b)"and "Proof(d)";when comparingTime
with
Time 4, all students'judgmentsimprovedor stayedthe same. However,the
1
datafor "Proof(a)"and"Proof(c)" aremorecomplex. Forexample,on "Proof(a)"
the improvementfrom four correctjudgmentsat Time 1 to five at Time 4 resulted
from two studentschangingto correctjudgmentsand one studentchangingto an
incorrectjudgment.
In general,the studentsconsiderablyimprovedtheirjudgmentsover time (even
though this was neither the purpose nor the expectation of the interviewer).
Furthermore,this improvementseems largelyto have been due to interactionswith
the interviewer that somehow enhanced the way that these students used the
experiences and knowledge that they brought to the task. This phenomenon
appears similar to occurrences in the zone of proximal development that were
describedby Vygotsky (1978) in studyingthe sociocognitivedevelopmentof children. It would be interestingto know more about such student-interviewerinteractions when the interviewer'spurposewas to enhancea student'smathematical
practices.Forexample, small portionsof these interviewsmightbe seen as dialectical situationsthatallowed the interviewerto suggest certainaspects of the practices and norms of the wider mathematicalcommunity. In other portions of the
interviews, a small comment or question apparentlyinduced a studentto refocus
his or her attentionon anotherpart of the task (i.e., on anotherpartof the argument being validated).
Whatthe StudentsSaid as TheyCarriedOut Their Validations
Most of the errors detected were of a local/detailed nature rather than a
global/structuralnature.For example, in validating"Proof(a)," five students(JB,
KC, SD, BH, ST) commentedon notationalerrorslike n2 = 9n2. Four (KC, SD,
MM, ST) noted that odd/even was not correctlyexpressed in the symbolism, for
example, 3n + 1 for odd. And two (BH, ST) observedthat divisibility by 3 could
not be concluded in the odd case from the expression 3n(n + 2) + 1.
AnnieSeldenandJohnSelden
25
Three students(KC, RC, MM) queriedthe interviewerregardingthe use of odd
and even cases in "Proof(a)," but they were willing to continue their validation
efforts aftermutuallyagreeingthat,althoughthis particularbreakdowninto cases
might be peculiar,nothing was wrong with it logically. For example, studentRC
read"Proof(a)" silently for 1 minuteandthenthe following interchangeoccurred.
is howcantheysay
RC: I didn't.... Itlookslikea proof.TheonlythingI'mquestioning
it's oddor even.Canyou do that?[It]wouldseemthat..... It seemslike it would
takecareof bothcases.... Thepositiveintegers....
I:
Wewerejustbeginning
to talkaboutthatinclass.... If youfeelyouwanttodosomeit's OK.
thingby casesin a proof,as longas thecasescovereverything,
RC: OK.
But
I:
Sometimeslikethis,they[meaningtheoddandevencases]arenonoverlapping.
Itdoesn'thurtto provesomethingtwiceaslongasthe
theycanevenbe overlapping.
unionof thecasescoverseverything.
RC: It'skindof tricky.I wouldneverhavethoughtof that.Whatever's
givenme,I tryto
proveit.
[RCcontinuesto considertheproofforsometime.]
I:
OKso, uh,if it's OKto do oddandeven,do youagreewithwhatthey'vedone?Is
thata proof?
RC: It lookslikeone. [Hethenexamines"Proof(b)."]
StudentBH, who had successfully proved the theoremearlierin the interview
aftera false start,thoughtthatthe conclusion in "Proof(a)" was being assumedin
the even case. That is, she apparentlyignored [4] and wrongly interpreted[5] to
includethe implicitsuppositionthatn = 3m. On readingfurther,she concludedthat
the conversehad been proved.She stated" ... [they're]doing it the way I wanted
to," suggesting thather initial failed attemptto prove the result influencedher to
incorrectlysee her own misstep in the argument.Only BH and ST had succeeded
in proving the theoremearlierin the interview;they were also the only students
who correctlystatedthatthe converse had been shown in "Proof(d)."These sorts
of structuralcomments were rare. Thus, as one might expect, the experience of
provinga theoremcan havepositiveeffects on subsequentvalidationsof otherarguments for the same theorem.And, perhapscounterintuitively,such earlierexperiences can sometimeshave negativeeffects on validations,as describedabove when
BH seems to have been more influencedby an early false starton a proof thanby
her later success.
Whatthe StudentsSaid TheyDid WhenReading Proofs
Most students seemed content to attempt a careful line-by-line check to see
whether each mathematicalassertion followed from previous assertions.When
queried (in Phase 4) about what they do when readinga proof, the studentssaid
they did manythings:They made sureall steps were logical andlooked to see that
everythingwas supported.They checked the computationsand whetheranything
was left out. They said, "[I] take it step by step; [I] like to read it real slow" and
"I go throughmore than once. ... See if from step to step, it follows." Several
26
as Texts
Validations
of ProofsConsidered
studentssaid thatthey would try to go througha proof with an example, but few
gave evidence of doing so duringtheir validations.
KC, whose validationof "Proof(d)"was paraphrasedabove, said, "[It's]not like
readinga newspaper;moreunderstandingis involved.I go throughmorethanonce.
Somethingwill come out at me thatdidn'tthe firsttime."Whenqueriedabouthow
she tells thata proof is not correct,KC explained,"becausethey didn't show it and
they didn't say it [meaningthe conclusion]."She added,"PlusI expect to see the
steps thatsay, thatlead you up to n is ... or whatever."She furtherexplained,"And
also if you find somethingthatgoes, ... that,that, .... that is definitely wrong in
the stepsabove.Like the one up here [pointingto one of the four"proofs"].So that's
not ... [a proof]."Given KC's limitedexperiencewith validatingproofs,otherthan
those in textbooks, this final remarkaboutfinding somethingdefinitely wrong in
just one step seems quitereasonable.However,in the odd case of "Proof(a),"there
are four mistakes thatare extraneousto obtainingn2 is divisible by 3 from itself.
(See the textualanalysisabove.)In general,the students'self-descriptionssounded
thoroughand competent.
MakingSense as a Criterionfor DeterminingCorrectnessof Proofs
In general,for these students,a feeling of understanding
or not-that is, of making
sense or not-seemed to be an importantcriterionwhen makingajudgment about
the correctnessof these four "proofs."For them, it seemed a questionof whether
the writtentext, togetherwiththeireffortsat comprehension,engendereda personal
feeling of understanding.For example, afterinitially readingthrough"Proof(a)"
slowly, JB concentratedon the last statement,n2 = 9n2, reiteratingseveral times
that it did not make sense. Then he said, "I'm saying I don't think this is a proof
because it [n2 = 9n2] doesn't make sense." After the interviewersuggested that
perhapsthe authorhad meant to write n2 = 9m2,JB concurredand continuedhis
validationattemptfor some time, finally deciding incorrectlyit was a proof "after
variablerenaming."Althoughthejudgmentwas incorrect,it is clearthatJB's main
focus was on making sense, perhapsresiding in the form of some conscious, but
unarticulated,feeling of understanding.
When KC first came to the gap in "Proof(c)," she rereadthe argumentseveral
times and tried to expandon it (on paper).When she could not, she explainedher
thinkingto the interviewer,as follows.
KC: MaybeI just don'tunderstand
theconceptof... I ... ,I understand
3 dividesn2
I understand
all thatstuff,butI don't... I don't... I understand
gettingfromhere
... [from]there[n2= 3x]to there[nn= 3x],but fromhere[nn= 3x]to here[31n]....
I:
Uh,huh.So whenyoutriedto fill it outyoucouldn'tbasically?
[KCagreesandsaysshecouldprobablygive anexample,butthatis nota proof.]
KC: I stilldon'tthinkthat,... thatis... [aproof].I don'tknow.I don'tthinkthatis...
[aproof].
I:
So that'snota proof?
KC: ButI maydecidelater... [reserving
therightto rethinktheargument].
AnnieSeldenandJohnSelden
27
At this point, a sense of not understandingmakes KC cautious about whether
the argumentis a proof.Anotherstudent,MM, in summingup herfinaljudgments,
wrote under "Proof (c)" that it was not a proof and "I don't understandthe
reasoning."Then, under"Proof(d)," she wrote an asteriskfollowed by "Proofthis one is the easiest for me to understand."However,this argumentis not a proof
of the theorem,but rathera proof of its converse. In her case, a feeling of understandingdid not mean the argumentwas correctlylinked (via a proof framework)
to the theorem.
The studentsalso indicated(in Phase 4) thatthey usually readproofsfor understanding.StudentBH, who was takingnumbertheoryconcurrently,said "It's got
to be truebut got to make sense. Lastnight I was lost 20 minutesin numbertheory.
I found somebody's thesis in the library.Just one step [was missing].... " This
comment suggests that the feeling of making sense, or of understanding,is a
matterof considerableimportanceto BH-important enough to search through
theses in the library.StudentKC, commentingon her previouscalculustextbook,
said, "Someof the proofs in the calc book were 'off the wall.' I had to sit andwrite
them up; [I had to] go home and see where the things were coming from."
For these students,understandingwas requiredof them when readingproofs.
When given a proof by an instructoror in a textbook, they quite reasonably
assumed that it was correct.Indeed, they probablyhad very little previouspractice in decidingwhether"proofs"arecorrect,thatis, in validation.Thus,theyappear
to have dependedon theirwiderpersonalexperiencesof understanding,which did
not always serve them well in validation.This unreliabilitymay occur because a
feeling of understandingitself and one's tendencyto rely on it are influencedand
sharpenedby experiencesin validation-experiences these studentslargelylacked.
Also, althoughthe studentsspoke of understandingquite freely, they apparently
did not always mean more or less the same thing by it. Their responses to questionsin Phase4, togetherwiththeirvalidationattempts,suggestthattheyhada range
of meanings for understandingfrom verifying shortcomputationsin a text using
their own proceduralknowledge to a feeling of understandingassociated with a
gestalt view of an entireargument.Perhapsthis variationis due to a generaldifficulty in negotiatingmeaningfor conceptssuchas understanding,which do not refer
to underlyingphysical objects (e.g., cars or basketballs).
CONCLUSIONSAND IMPLICATIONSFOR TEACHING
What studentssay about how they read proofs seems to be a poor indicatorof
whetherthey can actuallyvalidateproofs with reasonablereliability.They tendto
"talka good line." They say thatthey check proofs step by step, follow arguments
logically, generate examples, and make sure the ideas in a proof make sense.
However, their first readingjudgments(see Table 1, Time 1) yield no betterthan
chance results, suggesting thatthey cannotreliably implementtheir intentions.
On the other hand, even without explicit instruction,the reflection and reconsiderationengenderedby the interview process eventually yielded 81% correct
28
Validations
as Texts
of ProofsConsidered
judgments (see Table 1, Time 4). This improvementsuggests that instructionin
validationcould be effective. The interviewsalso suggest that studentsshouldbe
errors,forexample,proving
encouragedto attendmoreto possibleglobal/structural
the converseof the statement.In particular,studentsneedto devoteattentionto what
we have called proof frameworks(Selden & Selden, 1995), a topic thatwe suspect
is relatively easy to masterbecause it does not dependon a deep understandingof
the mathematicalconcepts involved in the statementof a theorem.
Validationof proofs is partof the implicit curriculum,but it is a largely invisible mentalprocess.Few universityteacherstryto teachit explicitly,althoughsome
may admonishstudentsto "readwith pencil and paperin hand"or the like. This
advice is at best descriptive,butcertainlynot usefullyprescriptive,so studentstend
to interpretsuchvaguedirectionsidiosyncratically,spottingmainlylocal notational
and computationaldifficulties. They appear to be substitutingtheir feeling of
overall understanding,plus a focus on surfacefeatures,for validation.We suspect
such a feeling of overall understandingis useful and reliable for mathematicians
who will have includedin it theirnotionof correctness,but is often unreliableand
misleading for students.
"Bridge"courses in the United States tend to give students some idea of the
logical structureof theorems,coveringsuch topics as directandindirectproofsand
indicatingthatthe converseis not logically equivalentto a theorem.These courses
also cover some logic, set theory, functions, and assortedothertopics. Often, the
materialon logic precedes studentwork with proofs, and perhapsfor thatreason,
tends to be presentedin an abstract,decontextualizedway that may not be very
effective. Textbooks for such courses have from none to just a few to a moderate
numberof exercises that involve the critiquingof "proofs."The directionsvary;
studentsmay be askedto (a) find the fallacy in a "proof';(b) tell whethera "proof'
is correct;(c) gradea "proof,"A for correct,C for partiallycorrect,or F; or (d) evaluatebotha "proof'anda "counterexample."
Most of these "proofs"havebeen carefully constructedby textbook authors so there is just one error to detect. For
example,in Smith,Eggen, andSt. Andre(1990; p. 39) we find the following "proof
to grade":
Suppose m is an integer.
Claim. If m2 is odd, then m is odd.
"Proof." Assume m is odd. Then m = 2k + 1 for some integerk. Therefore,m2
= (2k + 1)2= 4k2+ 4k + 1 = 2(2k2+ 2k) + 1, which is odd. Therefore,if m2is odd,
then m is odd. E
The "proof'above does not presenta studentwith anywherenearthe complexity
or kindsof difficultiesseen in our student-generated
Considerationof
arguments.'4
such contrived"proofs"is no doubtvaluablepracticefor those beginningto learn
validation.However, we suggest that additionalpracticein validatingsuch actual
14The Smith, Eggen, and St. Andretextbook (1990), and subsequenteditions thereof,does contain
more complex "proofsto grade,"however, many transitioncourse textbooksdo not.
AnnieSeldenandJohnSelden
29
student"proofs"as those in this study,togetherwith small-groupdiscussion,could
be beneficial,especially for preservicesecondaryteacherswho one day may need
tojudge the correctnessof theirown students'proofsor novel solutionsto problems.
FUTURERESEARCHQUESTIONS
One might consider the studentsin our empiricalstudy as novices at validation
and ask what experts-advanced undergraduates,graduatestudents,and mathematicians-would do. The studentsin our study could not reliably validaterelatively shortpurportedproofsof a simpletheorem,but mathematicianscan. At some
point, therefore,mathematicsstudentslearnto do so. When does this happen,and
throughwhat kinds of experiences is the process of validationlearned?
Proofs of the theorem used in this study do not seem to have a very complex
structure.What would happen if one considered a theorem whose proof was
inherentlymorecomplex or wherethe potentialfor moresubtleerrorswas greater?
Studentsoften have views of proof (Harel & Sowder, 1998) thatdiffer markedly
from mathematicians'views. How would such views be influenced by attempts
to increase students' ability to validate proofs?
How does the ability to validate proofs relate to the ability to constructthem?
How does it relate to a knowledge of logic, especially when taught in the rather
formal,symbolic, decontextualizedway thatis found in many transitioncourses?
What are students' perceived and actual criteria for correctness, for example,
thatthey agree the calculations are legitimate, thatthey can follow the argument
line by line, that they can see why the argumentbegins where it does, and so
on? The students in this study were sometimes misled by a feeling of overall
understanding, whereas mathematicians can use such feelings beneficially.
What accounts for the difference? How does one learn to have a feeling at the
"righttime" and not otherwise?Whatpartsof validationsconsist of readingplus
assent, without additionalconscious experiences, such as inner speech or vision
or feeling? What kind of teaching might promote a correct use of assent during
validation?
Here, after introducingthe idea of validation,we discussed its naturein a way
thatdependedon informationwe taketo be commonknowledgeamongmost mathematicians.However, withoutadditionalinvestigations,thereis no way to be sure
that we are right about what is common knowledge or that the common knowledge itself is correct.Taking much of this discussion as partlyconjecturalyields
a numberof additionalresearchquestions. For example, suppose one recorded
what was being attendedto during the constructionof a proof and extracted,in
order, those parts that appearedin its final form. Would this order, as we have
suggested, differ greatly from the orderof the final proof? In teaching, one of us
has sometimesencouragedstudentsto write a proof framework,thatis, the beginning andending few lines of a proof, before attemptingthe middle. Some students
resist this advice, and it would be interestingto investigatewhethersuch students
have more difficulty constructingproofs than others.
30
Validations
as Texts
of ProofsConsidered
We have also alludedto mathematicians'ability to agree on the correctnessof
proofs and the truthof theoremsand that they can often decide this individually
and believe "you do not vote on the truthof theorems."It would be interestingto
investigatethe degreeto which these arethe perceptionsof mathematiciansas well
as the degree to which those perceptions are accurate.Informationabout such
perceptions and abilities might enhance the role played by validation in social
constructivistviews of mathematics.
Last, what is a good way to teach validation?Is it a concept like proof or definition, which is perhaps best learned through experiences, such as finding and
discussing errorsin actualstudent"proofs"?
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Authors
Annie Selden, TennesseeTechnologicalUniversity,MathematicsDepartment,Box 5054, Cookeville,
TN 38505; selden@tntech.edu
John Selden, MathematicsEducation Resources Company, 1015 Melrose Drive, Cookeville, TN
38501; js9484@usit.net
APPENDIX 1
Facts Aboutthe Positive Integers
FACT 1. The positive integers,B can be dividedup into threekindsof integersthose of the form 3n for some integern, those of the form 3n + 1 for some integer
n, and those of the form 3n + 2 for some integern.
For example,
1,
2,
3,
4,
3n 3n+1
where n = 1
6,
5,
7,
3n+2 3n 3n+1
where n = 2
8,
3n+2
9,
10,
11...
FACT 2. Integersof the form3n (thatis, 3, 6, 9, 12, .. .) arecalled multiplesof 3.
FACT 3. No integercan be of two of these kinds simultaneously.So m is not a
multipleof 3 means the same as m is of the form 3n + 1 or 3n + 2.
APPENDIX 2
InterviewData Collection Phases
PHASE 1: Warm-UpExercises
For any positive integer n, if n2 is a multipleof 3, then n is a multipleof 3.
1. Explain, in your own words, what the above statementsays.
2. Give some examples of the above statement.
AnnieSeldenandJohnSelden
33
3. Does the above statementseem to be true?How do you tell?
4. Do you thinkyou could give a proof of the above statement?
PHASE2: Sequentialconsiderationof 'Proofs' (a), (b), (c), (d).
PHASE3: 'Recap' on the 'Proofs'
Below are several purportedproofs of the following statement:
For any positive integer n, if n2 is a multipleof 3, then n is a multipleof 3.
For each one, decide whetheror not it is a proof. Try to "thinkout loud" so you
can let me in on your decision process. If it is not a proof, point out which part(s)
are problematic.If you can, say where, or in what ways, the purportedproof has
gone wrong.
Assume thatn2 is an odd positive integerthatis divisible by 3. Thatis
(a). PROOF:
n2= (3n + 1)2= 9n2+ 6n + 1 = 3n(n+ 2) + 1. Therefore,n2is divisibleby 3. Assume
that n2 is even and a multiple of 3. That is n2 = (3n)2 = 9n2 = 3n(3n). Therefore,
n2 is a multipleof 3. If we factorn2 = 9n2, we get 3n(3n); which means thatn is a
multipleof 3. E
(b). PROOF:
Suppose to the contrarythatn is not a multipleof 3. We will let 3k be
a positive integerthatis a multipleof 3, so that3k + 1 and 3k + 2 are integersthat
are not multiplesof 3. Now n2 = (3k + 1)2= 9k2+ 6k + 1 = 3(3k2+ 2k) + 1. Since
3(3k2+ 2k) is a multipleof 3, 3(3k2+ 2k) + 1 is not. Now we will do the otherpossibility, 3k+ 2. So, n2= (3k+ 2)2 = 9k2+ 12k+ 4 = 3(3k2+ 4k + 1) + 1 is not a multiple
of 3. Because n2 is not a multipleof 3, we have a contradiction.U
Let n be an integersuch thatn2 = 3x wherex is any integer.Then 31n2
(c). PROOF:
Since n2= 3x, nn = 3x. Thus3In.Thereforeif n2is a multipleof 3, thenn is a multiple
of 3. M
Let n be a positive integersuch thatn2 is a multipleof 3. Then n = 3m
(d). PROOF:
where m e Z+. So n2= (3m)2= 9m2= 3(3m2).This breaksdown into 3m times 3m
which shows thatm is a multipleof 3. M
PHASE4. Final Questions
1. When you read a proof is there anythingdifferentyou do, say, than in reading
a newspaper?
2. Specifically, what do you do when you read a proof?
3. Do you check every step?
4. Do you read it more thanonce? How many times?
5. Do you make small subproofsor expand steps?
6. How do you tell when a proof is corrector incorrect?
7. How do you know a proof proves this theoreminsteadof some othertheorem?
8. Why do we have proofs?
Validationsof Proofs Consideredas Texts
34
APPENDIX 3
Transcript of KC Validating "Proof (d)"
[Note: Segmentation into stages is our own. Emphases in boldface indicate portions
of the transcript discussed earlier in the condensed version.]
Initial validation attempt in Phase 2
Stage 1
I:
OK, this is the last one. We have four of these.
KC: OK. [Reads theorem aloud slowly once. There's a pause at the end of the first
sentence, after which she says "OK"and goes on reading.]Yah. That, that looks a
lot like the one thatwe just did.
I:
Ahh, which means it's good or it's not good?
KC: Yah. No, notjust did. I mean this one over here [pointingto the paperwith (a)].
I:
Oh, it looks like. ... [Spreadsout the fourpaperswith the 'proofs'on them.] We had
(a), (b), (c), (d).
KC: Like thatone. Like thatone. [Pointingto (a).]
It looks like.... It looks a lot like the, ah, (a) one.
I:
KC: Yah.Yah,I thinkthat'sa proof.
Stage 2
OK.OK.You thinkthisone's a proof.Ookay.
I:
KC: 'Cause,see, 'causethis is whatI was talkingaboutby 3 times3m2.
I:
KC:
I:
KC:
Ah, OK.
Then you could let this 3m2be some integerk.
OK. So you want to see this displayedlike that?
Yah.
I:
OK.
KC: Then you could say thatlike...
I:
Well, I don't want to have it upside down [turnspaperaround].
KC: You could also ... say ... this breaksdown. Well, this one. ... WhatI would do, is
say, uhm, uhm 3 ... then, 3m2equals to k, some integerk such that3k, or something
like, equals to 3k, uhm....
I:
[Interrupting]Why don't you write thatdown? Your talkinghere.
KC: OK.
You need your pencil then.
I:
KC: Like here, I probablywould say [writes and talks more slowly]. Let 3m2equal to k,
for some positive integerk... [pause] ... integerk. OK. Uhm. [long pause] Then 3
times 3m would equal to 3k... [pause] which ... [pause]makes, uhm, or I don't
know the wordingmay have to be wrong,uhm, which makes,uhm ... [pause]which
is ... ,I don't know how to say it, but that's....
That's what you wantedto say.
I:
KC: See that's whatI wantto say, andthen, like, insertit in here.Then you could say that
this, by proving that it's three times some integer, some .... Then you could say
that it's now ...
I:
It's now a multipleof 3. [Finishes sentence for KC.]
Annie Selden and John Selden
35
Yah, it's now a multipleof 3, OK. [Says and writes] Now n2 is a multiple of 3.
Uh, huh.
I don't know if I'd word it in thatway, but that's what I'm thinking.
Thatn2 is a multiple of 3?
I don't know if I'd word it in thatway.
'Cause then you have it in the form three times something else.... Because this
breakingdown to 3m times 3m, uh that,thatkindof... becauseyou have this already.
I:
Uh, hmm.
KC: Do you understandwhat, what I'm saying?
KC:
I:
KC:
I:
KC:
KC:
Stage 3
I:
Yah. Yah. I understandyou want to get it displayedlike that.Uhm, and that's good
enough for a proof now?
KC: Now, my wordingrighthere probablyis wrong , I mean it could be....
I:
There's lots of differentways you can word things .... [Continuesfor some time.]
KC: I mean.I mean.I could. ... A lot of this could be like... Even this could be reworded.
You don't have to say "let"orjust say, you know, k, uhm, for some integerk equals,
or something,orjust let this be somethingelse. So you could have it in the form....
Or even that's fine. But, this 3m times 3m stuff, that's gotta go.
Stage 4
I:
Then, how do you know you're finished with? I mean....
KC: [Interrupts]
Because,becauseyou have it now, now you have it. ... [Longpause]Let's
see. They showed that m was a multiple of 3, but they didn't show that n was a
multiple of 3. [Long pause] 'Cause you have to show thatn is a multipleof 3.
Stage 5
I:
Uh, huh.
KC: Uhmmm.[Longpause] Hmmm.[Longpause] So you'd have to go back and you'd
have to substitute back in here. [Writes(3m)(3m)= (3m)n = 3(mn).]
I:
Uh, huh.
KC: Anyway that .... [Longpause]Well, it would. ... OK, wait. OK, wait, wait. .... Then
you'd have .... Let's see ... [long silence, lasting 30 seconds]. Well, [taps pencil
quicklytwice, andagainquicklytwice] ... [pause]I don't think that's a proof [said
definitely].
Stage6
I:
Oh! [surprised]Oh, OK. [KC laughs.] Now why did you change your mind?
KC: Well, because they didn't, they didn't say ... thatn was ... they didn't say n was a
multiple of 3. They didn't figure that out. They showed that m was a multiple of
3, but they didn't show that n was ... and....
Stage
I:
KC:
I:
7
[Interrupting]Where'd that m come from anyhow?
Well, the m came from.., they let n = 3m right up here [pointingto top of proof].
Yah.
KC: Justto, just to to...
I:
That'sjust like your using k, as far as I can tell.
36
Validationsof Proofs Consideredas Texts
KC: See, you're, and ... they're supposedto have let n2 be a multipleof 3, so it should
have been "let n2 = 3m."
Stage 8
I.
Uh, hmm.
KC: So. ... So, at leastI showedyou how I was, ... I wouldhave worded.... [Bothlaugh.]
I don't think that's a proof.
I:
OK this one's not a proof. OK, that's one's not a proof. OK, we can come back and
change our minds.
Subsequent "recap" validation attempt in Phase 3
KC: [Reads "proof' aloud, followed by a pause of 18 seconds.] No.
I:
Not a proof?
KC: No, because they still, I mean, they didn't define ... they didn't prove what the n
was, which is what they were supposedto prove.
I:
So they didn't prove what they were supposedto prove. Which was what?
KC: That,thatn is a multipleof 3.
I:
Uh, hmm.. . . OK.