Reducing Abstraction Level When Learning Abstract Algebra ...

Reducing Abstraction Level When Learning Abstract Algebra Concepts Author(s): Orit Hazzan Source: Educational Studies in Mathematics, Vol. 40, No. 1 (Oct., 1999), pp. 71-90 Published by: Springer Stable URL: http://www.jstor.org/stable/3483306 Accessed: 18-03-2015 12:19 UTC

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REDUCINGABSTRACTIONLEVELWHEN LEARNING ABSTRACTALGEBRACONCEPTS

ABSTRACT. How do undergraduatestudentscope with abstractalgebraconcepts?How should we go about researchingthis question? Based on interviews with undergraduate studentsand on writtenquestionnaires,a theoreticalframeworkevolved which could coherentlyaccount for most of the data. According to this theoreticalframework,students' responses can be interpretedas a result of reducingthe level of abstraction.In this paper, the theme of reducing abstractionis examined, based on three interpretationsfor levels of abstraction discussed in mathematicseducation researchliterature.From these three perspectiveson abstraction,ways in which studentsreduce abstractionlevel are analyzed and exemplified.

1. INTRODUCTION

The researchdescribedin this paperstrivesto accountfor mentalprocesses of undergraduatestudentsas they solve problemsin abstractalgebra.As it turnsout, mathematicseducationresearchliteraturebecomes increasingly scarcewhen one moves fromelementaryschool level, to high school level, and on to college level (cf. Selden and Selden, 1993; Thompson, 1993; Dreyfus, 1990, 1995). Even at college level, most of the researchdeals with pre-calculus,calculus, linear algebraand discrete mathematics(see Hazzan, 1995, for a comprehensiveliteraturesurvey). Only recently has serious researchbeen directedtowardslearning and teaching of abstract algebra.Paperscomprisingliteratureon abstractalgebralearningcan be roughlydividedinto two groups: a) Teachingmethods of abstractalgebra(Pedersen, 1972; Lesh, 1976; Macdonald, 1976; Buchthal, 1977; Quadling, 1978; Lichtenberg, 1981; Simmonds, 1982; Freedman,1983; Petricig, 1988; Thrasand Walls, 1991; Leronand Dubinsky,1995). b) Learning,understandingandconceptdevelopmentin abstractalgebra (Selden and Selden, 1987; Hart, 1994; Dubinsky,Dautermann,Leron and Zazkis, 1994; Hazzan, 1994; Leron, Hazzan and Zazkis, 1994, 1995; Hazzan and Leron, 1996; Brown, DeVries, Dubinsky and Thomas, 1997, Asiala, Dubinsky,Mathews,Morics and Oktag,1997; Asiala, Brown, KleimanandMathews, 1998). La

EducationalStudiesinMathematics 40: 71-90, 1999.

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The researchdescribedin this paperbelongs to the second category.Research findings presentedin Section 3 describe ways in which students deal with abstractalgebra concepts by making these concepts mentally accessible.More specifically,the ways in which studentsconceive abstract algebraconcepts are analyzedthroughthe theme of reducingthe level of abstraction.As it turnsout, in manycases, reducingthe level of abstraction is an effective strategy.However,sometimesit can be used inappropriately andbecomes misleading. The importanceof learningabstractalgebrais widely acknowledged. Gallian (1990) says that 'abstractalgebra is importantin the education of a mathematicallytrainedperson.The terminologyand methodologyof algebraareused evermorewidely in computerscience, physics, chemistry, and datacommunications,and of course, algebrastill has a centralrole in advancedmathematicsitself.' (p. xi). Instructorsare awareof the importance of learningabstractalgebra.At the same time, many of them report difficulties on the part of the students in understandingideas they (the instructors)try to communicate.Thus, educatorstry to find ways to help studentsunderstandabstractalgebraconcepts and look for ways, relevant for the students, to introducethese concepts. For example, in the introduction to his book AbstractAlgebra, Herstein (1986) argues that since '[t]here is little purpose served in studying some abstractalgebraobject without seeing some nontrivialconsequences of the study, [we present in the book] interesting,applicable,and significantresults in each of the systems we have chosen to discuss.' (p. viii). In a similar spirit, Gallian (1990) describes his approachfor presentingabstractalgebraideas in a book: 'WhatI have attemptedto do here is to capturethe traditionalspirit of abstractalgebrawhile giving it a concretecomputationalfoundationand includingapplications.I believe that studentswill best appreciatethe abstracttheorywhen they have a firmgraspof just what is being abstracted.' (p. xi). Gallian(ibid.) and Herstein(ibid.) referto the learningof abstract algebra in the traditionallecture hall. Nowadays, several initiatives use programminglanguages (such as ISETL or Maple) to teach abstractalgebra.For example,DubinskyandLeron(1994), who use ISETL,attempt 'to help createan environmentin which studentsconstruct,for themselves, mathematicalconcepts appropriateto understandingand solving problems in this area.'(p. xvii). Despite these attemptsto improvethe teachingof abstractalgebra,usually it is the first undergraduatemathematicalcourse in which students 'must go beyond learning 'imitativebehaviorpatterns'for mimickingthe solution of a large number of variationson a small number of themes (problems).'(Dubinskyet al., 1994, p. 268). Indeed, it is in the abstract

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algebracourse that studentsare asked, for the firsttime, to deal with concepts which are introducedabstractly.That is, concepts are defined and presentedby their propertiesand by an examinationof 'what facts can be determinedjust from [the properties]alone.' (Dubinsky and Leron, 1994, p. 42). The new mathematicalstyle of presentationleads students to adopt mental strategieswhich enable them to mentally cope with the new approachas well as with the new kind of mathematicalobjects. A plausible theoreticalframeworkto explain students'ways of thinkingin abstractalgebrasituationsis the focus of this paper.

2.

METHOD

As mentionedabove, the researcharea of understandingabstractalgebra studentsis relativelyuncharted.'Now the growconceptsby undergraduate ing body of researchliteratureconcerningthe learningof collegiate mathematicscontainsa few studiesfocusing on abstractalgebra.'(Asiala et al., 1998, p. 13). This trend is reflected mainly by Category (b) mentioned above. A specific frameworkfor cognitive research and curriculumdevelopmentof collegiate mathematicshas guided severalstudies from this category.This frameworkis developed and being used by RUMEC-1, a subgroupof the largerResearchin UndergraduateMathematicsEducation Community(RUMEC).A completediscussionof this researchframework is describedin Asiala et al., 1996. The researchdescribedin this paperexamines students'understanding of abstractalgebrafrom a differentperspective.In this research,the data haveguidedthe subsequenttheoreticalorganization,in the spiritof 'grounded theory' (Glaser and Strauss, 1967). Glaser and Straussexplain that '[g]eneratinga theoryfrom datameans thatmost hypothesesandconcepts not only come from the data,but are systematicallyworkedout in relation to the data duringthe course of the research.'(p. 6). This attitude,which intertwinesdevelopmentof a theory with the researchprocess itself, is especially suitablein cases such as the one describedin this paperin which a new field is investigated. In orderto gain a wide and variedpictureon which to base the theoretical framework,the datawere collected from a varietyof sources.Some of these sources were planned (interviews,written researchquestionnaires, regularclassroom tests, homeworkassignments),and others were incidental (observationsin abstractalgebra classes and occasional talks with students).Althoughnot all of these sources are explicitly presentedin the presentreport,they did help determinethe directionsand shape the ideas in the early stages of the study. This content downloaded from 158.121.247.60 on Wed, 18 Mar 2015 12:19:48 UTC All use subject to JSTOR Terms and Conditions

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Semi-structuredinterviewswere the main researchtool. The interview questionsappearin the Appendix.Nine undergraduatestudentsattending standardabstractalgebralectures, based on the 'theorem-proof'format, were interviewedduringtheirfirstcoursein abstractalgebra.1 The students were computerscience majorsin theirfreshmenyear in a top-rankIsraeli university.Prior to the abstractalgebracourse, the studentshad taken a linearalgebracourse, which had been taughtin a ratherabstractapproach, stressing proofs and abstractvector spaces. Additive and multiplicative arithmeticgroups, permutationgroups, and groups of symmetries of a regularpolygon were introducedas examples in lectures priorto the interviews.Each studenttook partin five weekly interviews,each interview lasting 1 hour.Questionsandstudentdatawere translatedfromthe original Hebrewfor this paper. The interviewsfocused on five fundamentalabstractalgebrasubjects: groups,subgroups,cosets, Lagrange'stheorem,andquotientgroups.Questions presentedin the interviewswere relativelysimple and mathematical knowledge requiredto solve them was relatively basic. Yet, in order to solve these questionsstudentshad to know the basic concepts,to examine relationshipsamong them, and to analyze their properties.For example, question24, 'Does a groupof even orderalwayshave a subgroupof order 2?', was chosen for its potential to reveal students' thinkingon groups, subgroups,and orders.At each stage duringthese five weeks, priorto the interview,the studentshad been introducedin the lectureto the concepts the interviewfocused on. Here are some additionaldetails aboutthe othertools used for collecting data:

* Writtenresearchquestionnaires:On severaloccasions,a writtenquestionnairewas distributedamong the membersof the whole class in orderto determinethe prevalenceof a certainphenomenonas it appearedin the interviews. * Regularclassroomtests and homeworkassignments:These tests and assignmentsreflected the expected requirementsof the course and were not specificallydesigned for the research.However,upon reading the work of the students,it was observedthat severalphenomena repeatedlyappeared. * Incidental discussions: Some students came to office hours to ask questions.These questionsoftenpointedto a student'sconceptualdifficulties and led to a brief impromptuinterviewon these difficulties. This content downloaded from 158.121.247.60 on Wed, 18 Mar 2015 12:19:48 UTC All use subject to JSTOR Terms and Conditions

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3. ABSTRACTION AND WAYS IN WHICH STUDENTS REDUCE IT

This sectiondescribesthe theoreticalframeworkdevelopedduringthe study, which aims to explain students'conceptionsof abstractalgebraconcepts. The title of this frameworkis reductionof the level of abstraction (in short, reducing abstraction).From this perspective,most responses and conceptionson the part of studentscan be attributedto their tendencyto work on a lower level of abstractionthan the one in which concepts are introducedin class. The term 'reducingabstraction'should not be conceived as a mental process which necessarily results in misconceptions or mathematicalerrors.The mental process of reducingabstractionlevel indicates that students find ways to cope with new concepts they learn. They make these concepts mentallyaccessible, so thatthey would be able to thinkwith them and handlethem cognitively. Abstractionis a complex conceptwhich has manyfaces in general,and in especially in the context of mathematicsand mathematicseducation.It has been dealt with by many psychologists and educators(e.g., Beth and Piaget, 1966). In the mathematicseducation researchcommunity it has been discussedfrom severalperspectives(cf. Tall, 1991; Noss andHoyles, 1996; Frorer,Hazzan and Manes, 1997). There is no consensus in regard to a unique meaning for abstraction.However,there is an agreementthat the notion of abstractioncan be examinedfrom variousperspectives,that certaintypes of concepts are more abstractthanothers,andthatthe ability 'to abstract'is an importantskill for carryingout meaningfulmathematics. Noss and Hoyles (1996) say: 'Thereis more thanone kind of abstraction.' (p. 49). Consequently,as will be discussed,thereis more thanone way to reducethe level of abstraction. The theme of reducing abstraction,presentedin this paper,is based on three interpretationsfor levels of abstraction discussed in literature. That is, abstractionlevel as the quality of the relationshipsbetween the object of thought and the thinkingperson, abstractionlevel as reflection of the process-objectduality,and abstractionlevel as the degree of complexity of the concept of thought.Relevantreferencesfor each interpretation appear later in the paper. It is importantto note that these interpretationsof abstractionare neitherindependentnor do they exhaust all possible interpretationsfor abstraction.As mentionedabove, abstraction is a complex concept and it is not the purposeof the paperto review all its interpretations.2 However,connectionsbetween the above threeinterpretations for abstractionare examined. In the following, the threeinterpretationsfor abstractionare discussed. Analyses of the ways in which studentsthink are examined in each case This content downloaded from 158.121.247.60 on Wed, 18 Mar 2015 12:19:48 UTC All use subject to JSTOR Terms and Conditions

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throughthe theme of reducing abstraction.In all cases, the reductionof abstractionlevel is a way of describingthe phenomenonthat in problemsolving situationsstudents(and in fact all problemsolvers) give the concepts involved some meaning. As it turnsout, the meaning studentsgive to some of the concepts can sometimesbe interpretedas being on a lower level of abstractionthanthatin which concepts are introducedin class. 3.1. Abstractionlevel as the qualityof the relationshipsbetweenthe object of thoughtand the thinkingperson This interpretationfor abstractionstems from Wilensky's (1991) assertion that whether something is abstractor concrete (or anythingon the continuumbetween those two poles), is not an inherentpropertyof the thing, 'but rathera propertyof a person's relationshipto an object.' (p. 198). In other words, for each concept and for each person we may observe a differentlevel of abstractionwhich reflects previous experiential connection between the two. The closer a person is to an object and the more connectionshe/she has formedto it, the more concrete(and the less abstract)he/she feels to it. Based on this perspective,some of students' mentalprocesses can be attributedto theirtendencyto make an unfamiliar idea more familiaror, in otherwords, to makethe abstractmore concrete. This idea is describedby Papert'smetaphorof looking for familiarfaces in an unfamiliarcrowd: For me, getting to know a domain of knowledge [...] is much like coming into a new communityof people. Sometimes one is initially overwhelmedby a bewilderingarrayof undifferentiatedfaces. Only graduallydo the individualsbegin to standout (Papert,1980, p. 137).

The idea of makingthe unfamiliarfamiliarin abstractalgebracontextis illustratedin this paperby students'tendencyto base arguments,whichrefer to generalgroups,on numbersand numberoperations- familiarmathematicalobjects,with which they have hadpreviousmathematicalexperience since elementaryschool. Selden and Selden (1987), suggest that '[e]ven [the] limited collection of examples [presentedat abstractalgebracourses at the junior level] is rich enough for studentsto see that not everything behavesthe way the morenaive studentsexpect.Whatthey expect is really a misconception,namely, that the rules they know for dealing with real numbersare universal.'(p. 462). Students'tendencyto rely on systems of numbers,when askedto solve problemsabout other groups, can be explainedby one of the basic ideas of constructivism.That is, that new knowledge is constructedbased on existingknowledge.Thus,unknown(hence abstract)objectsandstructures are constructedbased on existing mentalstructures.More specifically,if a This content downloaded from 158.121.247.60 on Wed, 18 Mar 2015 12:19:48 UTC All use subject to JSTOR Terms and Conditions

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given problemrequiresthe manipulationof mental objects (like groups) the studentshaven't yet constructed,they would tend to think with familiar objects (like numbers),and to solve the problemusing these familiar entities instead. The following two examplesillustratehow studentsreduce abstraction level by basing theirargumentson mathematicalentities they are familiar with - numbersand numberoperations. Example1: Groupsof numbers(such as Z in relationto additionor R-{0} in relation to numbermultiplication)are only some of the examples studentscome across in standardabstractalgebra courses. Since each of these groups possesses many additionalmathematicalpropertiessuch as commutativity and ordering,none is a typical example of a general group. Generic examples of groups,which studentscome acrossin abstractalgebracourses, are permutationgroups and groups of symmetries of regularpolygons. From the data analysis it turns out that studentsoften treat groups as if they were made only of numbersand of operationsdefinedon numbers. In the following excerpt,Tamartries to determinewhetherZ3 (i.e., the set {0, 1, 2} with additionmod 3) is a group. She takes for grantedthat 'the inverse' of 2 is 1/2 (as with rationalnumbersin relation to number multiplication),and tries to locate it in Z3. Since she does not find 1/2 in Z3 she concludesthatZ3 is not a group: 1Z3]is not a group. Again I will not have the inverse. I will not have one half. I mean, 1 over... I mean, if I define this [Z3] with multiplication,I will not have the inversefor each element. [ ...] It will not be in the set. [.. .] I'm tryingto follow the definition.[ ...] WhatI mean is thatI know thatI have the identity,1. WhatI have to check is if I have the inverse of each... I mean, I have to see whetherI have the inverseof 2 and I know thatthe inverse of 2 is 1/2- [.. .] Now, one half, and on top of thatdo mod 3 [ ...], then it is not included.I mean, I don't have the inverseof ... My inverse is not includedin the set. Then it's not a group.I do not have the inversefor each element.

In this excerptTamarautomaticallyconsidersthe groupoperationas number multiplicationand thus changes the operationfrom a less familiarone (additionmod 3) to a more familiarone. This may be a result of the use of the term 'multiplication'for a general binary operationin the group definition,togetherwith the need, mentionedabove, to mentallyhang on to some familiarentity.However,it is worthwhileto notice thatTamaruses the abstracttheoretic-terms(identity,inverse)correctly,and since a single axiom failuredenies a group,her explanation(hadto be takeninto account in respectto the groupoperationshe considered)makes sense.

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Example2: In many cases, the use of surfaceclues may help greatlyin problemsolving. But, the conclusionsthus reachedmust be carefullychecked.If this is not done, the tendencyto use surfaceclues may be quite misleading,as in the following misuse of Lagrange'stheorem. In a writtenquestionnaire(but not in the interviews), 113 abstractalgebrastudentswere presentedwith the following question: In an exam a studentwrote: 'Z3 is a subgroupof Z6'. In your opinion, is this statementtrue,partiallytrue,or false? Please explain your answer.

Incorrectanswers('the statementis true')were given by 73 students,20 of whom invokedLagrange'stheorem,in essentiallythe following manner: Accordingto Lagrange'stheoremZ3 is a subgroupof Z6 because 3 divides 6.3

By giving this answer,studentsnot only confuse Lagrange'stheoremwith its converse (If k I o(G), then there exists in G a subgroup of order k - which is not a true statement),but actually use an incorrect version of the converse statement(If o(H) I o(G) then H is a subgroup of G). Thus, a combinationof severallogic-basedconfusions, studentshave with mathematicaldeductions,appearto exist here. A detailedanalysis of this phenomenonis given in Hazzanand Leron(1996). However, in the context of this paper, I suggest that the popularity of Lagrange'stheorem in this context is derived from its strong link to mathematicalentities studentsare familiarwith, i.e., numbersand number divisibility.At the stagethe problemwas presentedto the students,manyof them felt mentallydisconnectedfromthe notionsappearingin the question (the groups Z3 and Z6, and the concept of subgroup).At the same time, the numbers3 and 6, were familiarand meaningfulto the studentsand, moreover,studentscould easily finda relationshipbetweenthe two, which led to a sensible solution to the problem.In short, studentswere guided by the appearanceof the numbers3 and 6 and the fact that 3 divides 6, ignoring the other details in the problem.From the point of view of the interpretationof abstractiondiscussedin this section,it seems thatstudents hang onto familiarmathematicalentities,ignoringthe meaningof the situation describedin the problem.It helps them work and think with more familiarconceptsor, in otherwords,makethe abstractmore concrete;that is, reducingthe level of abstraction. There seems to be a similaritywith findings about elementaryschool pupils solving 'story problems' (Nesher and Teubal, 1975; Nesher, 1980; Schoenfeld, 1985). Elementaryschool pupils often look for two numbers and for a word that will clue them as to whetherthe operationinvolved is additionor subtraction.Then they go on giving the solutionby performing This content downloaded from 158.121.247.60 on Wed, 18 Mar 2015 12:19:48 UTC All use subject to JSTOR Terms and Conditions

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the operationon these numbers,without even looking at the rest of the story.

3.2. Abstractionlevel as reflectionof theprocess-objectduality In this section the idea of reducing abstractionis reviewed based on the process-objectduality,suggestedby some theoriesof conceptdevelopment in mathematicseducation(Beth and Piaget, 1966; Dubinsky,1991; Sfard, 1991, 1992; Thompson,1985). Some of these theories,such as the APOS (action, process, object and scheme) theory, suggest a more elaborated hierarchy(cf. Dubinsky, 1991). However,for the discussion in this paper it is sufficientto focus on the process-objectduality. Theoriesthat discuss this duality distinguishbetweenprocess conception andobject conceptionof mathematicalnotions. Dubinsky(1991) captures the passage from the first conception to the second one as a 'conversion of a (dynamic)process into a (static) object'. Process conception implies thatone regardsa mathematicalconcept 'as a potentialratherthan an actualentity,which comes into existence upon requestin a sequenceof actions.'(Sfard,1991, p. 4). Whenone conceives of a mathematicalnotion as an object,this notionis capturedas one 'solid' entity.Thus,it is possible to examine it from variouspoints of view, to analyze its relationshipsto other mathematicalnotions and to apply operationson it. According to these theories,when a mathematicalconcept is learned,its conceptionas a process precedes- and is less abstractthan- its conceptionas an object (Sfard, 1991, p. 10). Thus, process conceptionof a mathematicalconcept can be interpretedas on a lower level of abstractionthat its conceptionas an object;thatis, abstractionlevel is reduced. The mental process that leads from process conceptionto object conceptionis one mechanismPiaget namedreflectiveabstraction.Dubinsky's work (1991) tells us aboutthe importanceof this mechanismfor mathematicalthinking:'Piagetconsideredthatit is reflectiveabstractionin its most advancedform that leads to the kind of mathematicalthinkingby which form or process is separatedfrom content and that processes themselves are converted,in the mind of the mathematician,to objects of content.'(p. 98). This perspective of levels of abstractionis related to (and may even arise from) the previousinterpretationfor abstractiondescribedin Section 3.1 - abstractionlevel as the qualityof the relationshipsbetweenthe object of thoughtand the thinkingperson.The more one works with an unfamiliar concept initially being conceived as a process, the morefamiliar one becomes with it and may proceedtowardits conceptionas an object. This content downloaded from 158.121.247.60 on Wed, 18 Mar 2015 12:19:48 UTC All use subject to JSTOR Terms and Conditions

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Several studies analyze the understandingof abstractalgebraconcepts from the more elaboratedperspective,that is, the APOS theory.For example,Brownet al. (1997) analyzethe understandingof the notionsbinary operations,groupsand subgroups;Asiala et al. (1997) presentan analysis of students'understandingof cosets, normalityand quotientgroups;Asiala et al. (1998) focus on the developmentof students'understandingof permutationsand symmetries. My contributionto the discussion is with two additionalaspects of process conception: (a) students' personalizationof formal expressions and logical argumentsby using first-personlanguage, and (b) students' tendencyto workwith canonicalproceduresin problemsolving situations. In the following examples,I explainhow these aspectsreflectprocessconception, and how they can be interpretedas mentalprocesses of reducing abstraction. Example3: This example illustrateshow studentspersonalizeformalexpressionsand logical argumentsby using first-personlanguage. It furthersuggests that this way of speakingreflectsprocess conception. The use of firstpersonlanguageis especially pronouncedin quantified expressions,where studentsoften replace 'thereexists' with 'I can find'. This may be explainedby difficultiesstudentsexperiencewith quantifiers (Dubinsky,Eltermanand Gong, 1988; Harnik, 1986; Leron, Hazzan and Zazkis, 1994; Dubinsky, 1997), and may also be related to the fact that quantifiersrarelyexist in high school mathematics(pun intended).Here, Dan, for example, explains how he checks the existence of inverses in question3 (see Appendix): Dan: I want to check if I can find for each... If for each element in the set I can find one element in the set I.. .1I want tofind an inverseso thatif I multiply,I will get e. (emphasis added).

The use of first-personlanguagereflectsa feeling of 'me doing something' and thus may be interpretedas process conception of the concepts the studentsthinkwith. In the spiritof reducingabstraction,it is suggestedthat the studentsapplymentalstrategythatmakesthe unfamiliarmathematical languagemore familiarfor them. Indeed, 3,500 years of the development of mathematicsare intertwinedwith first-personlanguageand recipes (cf. Kleiner, 1991). Thus, if studentswere asked in a one-semestercourse to conceive concepts that have been organizedinto a coherenttheory over long periodsof time, they (the students)have to find ways to cope with the unfamiliarterminologythey face. This content downloaded from 158.121.247.60 on Wed, 18 Mar 2015 12:19:48 UTC All use subject to JSTOR Terms and Conditions

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Here is one more example: Commutativityis a propertyof a set and of an operationdefined on the set elements. When one presents a statement such as: 'Operation0 is commutativefor all subsets of a set A', we may say that commutativityis conceived as an object. In the following excerpt, in contrast,Adam defines (non)commutativityby describinghis own actions: when I replace the place of the Adam: Commutativity- that means, that if I ... elements in the product then I get a different result. When I change the order in the multiplicationI get a differentresult, so they will not be commutative.(emphasisadded).

The use of firstpersonlanguageoften goes togetherwith very detailedand specific descriptions.In the following excerpt for example, Tanyaexamines the definitionof quotientgroupG/H = { Hx Ix in G} in her notebook and explains it: Tanya:I take all the elements [of H] and multiply them on the right with some element from G. I choose an element from G and I multiplyall the elements of H on the right. [ .. I I mean, I should do it for all... I do not do it together. [...] Each one by itself [...] So, I have to choose a andx. (emphasisadded).

Tanya'sdescriptionreflects process conception of a quotient group: she describes how she builds the cosets one by one, and how each coset is constructedby a sequence of products.Such a descriptionenables her to know exactly where she is in the process of the construction,which coset she has alreadybuilt and which ones are still ahead.By following this path studentsdo not have to capturethe whole situationat a glance as a single entity - hence they are not obliged to adopt an object conception. From the process-objectduality point of view, this interpretationsuggests that the level of abstractionis reduced. Example4: This example illustratesstudents' tendency to work with canonical proceduresin problemsolving situations.This tendencycan be explainedby the fact that some of the concepts, and the relationshipsamong them, are conceived as processes. The term canonical procedure refers to a procedurethat is more or less automaticallytriggeredby a given problem. This can happen either because the procedureis naturallysuggestedby the natureof the problem, or because priortraininghas firmlylinked a specific kind of problemwith a specific procedure.The availabilityof a canonicalprocedureenables studentsto solve problemswithoutanalyzingpropertiesof mathematicalconcepts, andto automaticallyfollow the step-by-stepalgorithmthe canonical procedureprovides. In the following quote, for example, Guy presents a This content downloaded from 158.121.247.60 on Wed, 18 Mar 2015 12:19:48 UTC All use subject to JSTOR Terms and Conditions

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clear analysis of the structureof the cosets in a group, but then 'forgets' to use that knowledge;in the solution of anotherproblem,he chooses the alternativeof a messy coset calculation.Here is his description: Guy: OK, it is knownthatequivalentclasses divide the set into disjointsets, the whole set. And now, if a coset is like an equivalentclass thenthe cosets divide the ... divide the group into disjointsets. We know that in each coset the numberof elements is equal.

In spite of this clear description,when asked to calculatethe cosets of the subgroup{ 1, 2, 4 } in Z7,-{ 0 } with multiplicationmod 7, Guy firstcalculated all six cosets (one for each element of the group).It was only when he noticed that there were just two differentcosets, that he began to look for the reason. Based on his clear descriptionof the cosets as equivalent classes with the same numberof elements, it is reasonableto assume that hadhe stoppedforjust a minutebeforejumpinginto a calculationmode, in orderto try to examine the objects in the question,he would have known right away that there were only two differentcosets.4 However,Guy did not use his theoreticalknowledge as a tool; instead,his firsttendencywas to start calculatingthe cosets by using a well-known procedurefor this kind of task. This immediateresponse was done unconsciously:Guy did not considerthe two alternatives(a use of the fact thatcosets areequivalent classes vs. a use of a canonicalprocedure),choosing to use the canonical procedure.It is moreplausibleto assumethatGuy unconsciouslyfollowed the paththatthe canonicalprocedureoutlinedhim. This example suggests that even in cases where studentshave a relatively advancedconceptionof mathematicalnotions, sometimesthey tend to use a (less abstract)canonicalprocedureat the cost of manycalculations. Indeed,it is mucheasierto make sense of a canonicalprocedurethanof an abstractargument,which usuallycapturesin one sentenceseveralconcepts togetherwith relationshipsamong them. Here, we can see that still, there is a gap betweenobjectconceptionof separatedconcepts andthe abilityto mentallylink them all togetherinto one schemaconceived as an object. 3.3. Abstractionlevel as the degree of complexityof the conceptof thought This section examinesabstractionby the degreeof complexityof mathematical concepts. For example, a set of groups is a more compoundobject thanone specific groupin thatset. Thus,the set of additivegroupsof prime orderis a more compoundmathematicalentity than the group [Z5,+5]. It does not imply automatically,of course, thatit shouldbe more difficultto thinkin terms of compoundobjects. The workingassumptionhere is that the more compoundan entityis, the more abstractit is. In this respect,this section focuses on how studentsreduceabstractionlevel by replacinga set This content downloaded from 158.121.247.60 on Wed, 18 Mar 2015 12:19:48 UTC All use subject to JSTOR Terms and Conditions

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with one of its elements,thus, workingwith a less compoundobject.As it turnsout, this is a handytool when one is requiredto deal with compound objects thathaven'tyet been fully constructedin one's mind. There is a connection between the interpretationfor abstractionsuggested in Section 3.2 (thatis, abstractionas reflectionof the process-object duality) and the one suggested here. This connectionties the set concept together with object conception and process conception: When the set concept is conceived as an object, a person becomes capable of thinking about it as a whole 'without feeling an urge to go into details.' (Sfard, 1991, p. 19); when conceived as a process, one conceives the set concept as a process in which its elements are grouped.Thus, when one deals with the elements of a set insteadof with the set itself we may interpretthis as process conceptionof the set concept. The following exampleillustratesthe tendencyto deal with an element in a set instead of with the whole set. More specifically,it describeshow studentsdeal with a specific groupwhen askedto deal with a set of groups containingit. Example5: This example describes a studentwho faced a problemconcerninga set of groups, and turnedto consider only one groupfrom the whole set (cf. also Rumelhart,1989, p. 301: reasoningby example). In other words, he replaceda set with one of its elements. Consideringa specific case may be a positive and helpful heuristic, as recommendedby Polya (1973): 'Specializationis passing from the considerationof a given set of objects to that of a smaller set, or of just one object, containedin the given set. Specializationis often useful in the solution of problems.'(p. 190). The role of such specializationis to guide towardsa generalsolution.However, therearecases in which studentsdo use this heuristic,presentingan answer based on an analysis of a specific case, and do not returnto the general case. Sometimes they do not go back to the general case simply because they are unable to do that- the mental structuresneeded to deal with the generalcase have not yet been constructed. Question24 in the interviewasks: 'Does a groupof even orderalways have a subgroupof order2?'. In the following excerptRon describeshow he 'justmisses the point' andthus turnsto work with a specific group.Apparently,he encountersdifficultiesin thinkingaboutthe family of groups of even orderor difficultiesin thinkingabouta genericgroupof even order. Ron: A group of even order... does it always have a subgroupof order 2? [...] [pause] Why yes or why no? I just feel thatI miss the point. [ .. .] A groupof even ordercan also be a group... let's say of order6 for example. A groupof even order?[ ...] Yes. If the order is 6 so it can have subgroups[of order]either2 or 3, of order2 or order3.

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4. SUMMARY This paper presents various mental processes by which studentsreduce abstractionlevel in abstractalgebraproblem-solvingsituations.As it turns out, in many cases, the mental mechanismof reducingabstractionhelps studentsto cope successfullywith problemspresentedto them.This theoretical frameworkcorrespondswith constructivisttheories (cf. Kilpatrick, 1987; Sinclair, 1987; Davis, Maher and Nodding, 1990; Confrey, 1990; Smith, diSessa and Roschelle, 1993) and with the Piagetianterminology of assimilationand accommodation(Piaget, 1977). Given the abstraction level in which abstractalgebraconceptsareusuallypresentedto studentsin lectures,and the lack of time for activitieswhich may help studentsgrasp these concepts,manyof the studentsfail in constructingmentalobjectsfor the new ideas and in assimilatingthem with theirexisting knowledge.The mentalmechanismof reducingthe level of abstractionenables studentsto base their understandingon their currentknowledge, and to proceed towardsmental constructionof mathematicalconcepts conceived on higher level of abstraction.

ACKNOWLEDGEMENT

This work is based on the Ph.D. thesis of the author,writtenunder the supervisionof Prof. Uri Leron.I would like to expressmy gratitudeto Uri Leron and Rina Zazkis for their useful remarkson an earlierdraftof this paper.

APPENDIX: THE INTERVIEW

Remarks: a) The following questionsservedonly as a guide to the interviewerand were presentedto the studentsorally. b) The questionsare groupedhere in five partsaccordingto theirfocus. However,the topics were not told to the studentsandthe passagefrom one topic to the following one was done withoutany declarationabout the currenttopic underthe discussion. Part 1: The concept of group (questions 1-6)

1. Whatis a group? This content downloaded from 158.121.247.60 on Wed, 18 Mar 2015 12:19:48 UTC All use subject to JSTOR Terms and Conditions

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2. Can you give an example of a commutativegroup and of a noncommutativegroup? 3. Does the set {a, b, c} form a grouprelativeto the following operation table? * a b c a c a b b a b c c b c a 4. a) Is the identityelement in a groupunique? b) Whatdoes it mean thatthe identityelementis unique? c) Prove thatthe identityelement in a groupis unique. 5. Does a groupwith only one elementexist? 6. Let's constructan operationtable of four elements a, b, c, d. Please fill it in such a way that it will form an operationtable of a groupof order4. Part 2: Theconcept of subgroup(questions7-17) 7. Whatis a subgroup? 8. Can you give an exampleof a group G and a subgroupof G? 9. Considerthegroups(Z3, +3 (Z6, +6). Is Z3 a subgroupof Z6? 10. Can you give an exampleof a subgroupof Z6? 11. How do you check thatnonemptysubsetof a groupis a subgroup? 12. Please prove thatthe example of subgroupyou gave in question8, is indeed a subgroup. 13. Considera groupG and define the following subsetof G: C = {a in G I1/ x in G, ax = xa}. Can you describeC's elements? 14. The set C definedin question#13 is called the center of the groupG. How would you check if a groupelement is (or is not) in the center of the group? 15. How will you check whether3 is in the centerof the group [Z, +]? 16. Whatis the centerof a commutativegroup? 17. Prove thatthe centerof a groupG is a subgroupof G. Part 3: The concept of coset (questions18-21) 18. Whatis a coset? 19. Whatis the numberof elements in every coset? 20. Considerthe group [Z7-{ O}, *7] and its subgroup[{ 1, 2, 4}, What are the cosets of the subgroupin the group? 21. What are the cosets of 2Z in Z?

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Part 4: Lagrange'stheorem(questions22-26) 22. 23. 24. 25. 26.

Does [Z6, +6] have a subgroupof order4? Does a groupof order 12 always have a subgroupof order6? Does a groupof even orderalways have a subgroupof order2? Can a groupof order7 have an elementof order4? a) How manynon trivialsubgroupsdoes a groupof primeorderhave? b) Can you prove your last claim?

Part 5: Theconceptof quotientgroup(questions27-30) 27. a) Whatis a quotientgroup? b) When does a quotientgroupexist? c) Can you state a propertyof a group that guaranteesthat all its subgroupsare normal? 28. Is the following statementtrueor false? If G is a group and H is a commutativesubgroupof G, then H is a normalsubgroupof G. 29. LetG be [Z6, +6] andH its subgroup[{ 0, 3 }, +6]. a) Is H a normalsubgroupof G? b) Whatare the elements of the quotientgroupG/H? c) What is the operationdefinedon the quotientgroup?(How do we multiplytwo elementsin the quotientgroup?) d) Can you constructthe operationtable of G/H? e) Does this groupremindyou of anothergroup? 30. Whatis the identityelementin a quotientgroup?

NOTES 1. As mentioned before, in the last several years there has been a trend (especially in the USA) to teach abstractalgebra with the programminglanguage ISETL (Dubinsky and Leron, 1994; Leron and Dubinsky, 1995). Understandingof abstractalgebra concepts by studentswho learnedthroughthis method are describedin the following papers:Brown et al., 1997, Asiala et al., 1997; Asiala et al., 1998. I also experienced the teaching of abstractalgebrathroughthis method (togetherwith Prof. Uri Leron) during 1993-1994. However, my intention in the study presentedin this paper was to constructa theoreticalframeworkto describethe understandingof abstractalgebra concepts by studentswho attendstandardlectures. 2. For example, there is the association of 'general' with 'abstract'.However, I agree with Stauband Stem (1997) who suggest that '[t]he degree of generalitywith respect to the range of (potential)real world references is one aspect of the abstractnature of quantitativeschemata;but, [...] generalityis not the core aspect that makes mathematicalconstructsespecially abstract'(p. 65). Similarly,Leron(1987) says that '[i]n This content downloaded from 158.121.247.60 on Wed, 18 Mar 2015 12:19:48 UTC All use subject to JSTOR Terms and Conditions

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mathematics,abstractionis closely relatedto generalization,but each can also occur withoutthe other'. 3. Lagrange'stheorem states that in a finite group the order of a subgroupdivides the order of the group. (The order of a group is the number- finite or infinite - of its elements.) A sophisticatedanswermight claim thatZ3 is a subgroupof Z6 because it is isomorphicto a subgroupof Z6 (i.e., {0, 2, 4} with additionmod 6). 4. Actually,since thereareonly two distinctcosets, thereis no need to do any calculations whatever:one coset must be the subgroup{ 1, 2, 4} itself, and the other one must be its complement{3, 5, 6}.

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Departmentof Science Education- Technion, Israel Instituteof Technology, Haifa 32000, Israel E-mail: [email protected]

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