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High school students’ perceptions ofgeometrical proofs proving and refutinggeometrical claims of the ‘for every … ’and ‘there exists’ typeDorit Patkin aa Kibbutzim College of Education, Mathematics Education, 149Namir Road, Tel-Aviv, Israel

Available online: 08 Mar 2012

To cite this article: Dorit Patkin (2012): High school students’ perceptions of geometrical proofsproving and refuting geometrical claims of the ‘for every … ’ and ‘there exists’ type, InternationalJournal of Mathematical Education in Science and Technology, DOI:10.1080/0020739X.2012.662301

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International Journal of Mathematical Education inScience and Technology, 2012, 1–14, iFirst

High school students’ perceptions of geometrical proofs

proving and refuting geometrical claims of the

‘for every… ’ and ‘there exists’ type

Dorit Patkin*

Kibbutzim College of Education, Mathematics Education, 149 Namir Road,Tel-Aviv, Israel

(Received 16 September 2011)

Proving and refuting mathematical claims constitute a significant elementin the development of deductive thinking. These issues are mainly studiedduring geometry lessons and very little (if at all) in lessons of othermathematical disciplines. This study deals with high school students’perceptions of proofs in the geometry. The study explores whether studentsknow when to use a deductive proof and when an example is sufficient forproving or refuting geometrical claims. The findings indicate that in casesof simple claims, the students corroborate them by using a deductive proof.However, when the claim is more complex, the students tend to presentboth a proof and an example. Moreover, they are unsure whether using anexample can constitute a method for proving a mathematical claim,believing that in mathematics everything must be proven. They believe thatexamples are used merely for illustration purposes rather than as a meansof convincing. The research conclusions support the need for deepeningand developing the students’ distinction between cases where examples areinsufficient and cases where an example is sufficient for proving a claim.

Keywords: deductive thinking; ‘for every . . . ’ claim; inductive thinking;proof; refutation; ‘there exists . . . ’ claim

1. Introduction

Proving claims in mathematics, refuting and using them are important tools inbuilding a mathematical system [1,2]. Moreover, the document ‘Principles andstandards for school mathematics’ [3] has attributed a central place to the role ofproofs. The document indicates that the concept of proof must be part of thelearners’ accumulated experience, starting from kindergarten until graduation:‘Reasoning and proof should be a consistent part of students’ mathematicalexperience in pre-kindergarten through grade 12’ (p. 56). The document also specifiesthat competences associated with proofs should be fostered in all the studiedcontents, not only in geometry. A mathematical proof is an expression of deductivethinking; however, sometimes, inductive thinking can assist learners to make a moreformal claim [4]. Hence, it is essential to use proofs that do not only ‘prove’ but also‘explain’ [5] as well as use various types of proof as a main working tool of

*Email: patkin@netvision.net.il

ISSN 0020–739X print/ISSN 1464–5211 online

� 2012 Taylor & Francis

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explanations, clarifying why the proofs are correct. This study is focused on highschool students’ perceptions of geometrical proof. Following a brief theoreticalbackground, the paper presents the ways students implement in order to cope withtrue and false claims. Moreover, it deals with the profile of the thinking of fivestudents.

2. Theoretical framework

2.1. Proofs and types of thinking: Induction and deduction

Induction is a way of thinking, whereby we convince ourselves that a certain proof iscorrect, based on individual cases. Deduction is a thinking process that is theopposite of inductive thinking. When using this method, the certitude of a claim isdetermined by a set of assumptions and inferences resulting from them, arranged ina logical structure of a proof devoid of internal contradictions. A deductive proof isa logical sequence of claims leading to a conclusion grounded only on terms andfundamental assumptions (that do not need proving) as well as conclusions drawnearlier on the basis of the terms and fundamental assumptions. Deduction is theprocess of deriving the consequences of what is assumed. Given the truth of theassumptions, a valid deduction guarantees the truth of the conclusion. A validargument in which the conclusion follows from the premises.

Studies show that students should undergo a thinking process before being ableto write a deductive proof. Eco [6] and Peirce [7] built models for describing thisthinking process. The models are based on a preliminary stage in the process ofthinking that is based on the presentation of premises and ideas. As Eco said: ‘Thereal problem is how to figure out both the rule and case at the same time, since theyare inversely related, tied together’ (p. 230).

Deductive thinking is the way by which generalization is performed. The abilityto make a deductive claim requires internalization of a structural scheme. Twoprerequisites must exist for students to acquire a certain structural scheme [8].

(1) Learners must attain an appropriate level of cognitive maturity.(2) Learners must undergo a process of suitable experiencing.

Six-year-old children are not yet cognitively mature to understand the concept of‘formal mathematical proof’. Consequently, in order to know whether the claim iscorrect or incorrect, they must undergo a suitable practical experience: explanations,examples and exercises.

Wohlgemuth [2] maintains that the ability to prove and assess a proof is the laststage in the hierarchy of proofs, whereas inductive thinking appears in earlier stages.He argues that it is impossible to reach advanced stages without having passed and‘matured’ (for a long time even) at the initial stages of the hierarchy. This is manifestedby the fact that some students are capable of making conjectures based on observationmade along the years, but are unable to logically validate their conjectures.

2.2. Claims in mathematics presented to students and their methodsof proof or refutation

The teaching of mathematics in the upper grades of high school focuses on two typesof claims: ‘for every . . . ’ and ‘there exists . . . ’. For example, the claim that every

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isosceles triangle has two equal angles is of the ‘for every . . . ’ type. These proofs aremade either formally (by means of a final system of conjectures and conclusionsderived from the conjectures) or by counting all the possible cases in order to ensurethe certitude of the claim. Refuting a ‘for every . . . ’ type of claim can also beperformed formally; however, it is usually done by means of a counterexample of‘non-existence’ in order to refute the claim. Using a (single) counterexample to refutea ‘for every . . . ’ claim is linked to knowing that the negation of a universalquantification is an existential quantification. For instance, in order to refute theclaim that ‘every two triangles with equal area are congruent’, one can draw twotriangles, one obtuse angled and the other acute angled, which are equal in one sideand in the altitude to this side and are non-congruent.

The second type is ‘there exists . . . ’ claim, as in the claim: there exists aparallelogram whose sides are all equal. Proving this claim is usually done bypresenting a supporting example, for example, presenting a rhombus as aparallelogram whose sides are all equal. For example: for the claim ‘there existtwo different parallelograms whose area is equal’ one can build two parallelogramsthat are not equal but one of their sides and the altitude to this side are equal.Examples are insufficient for refuting a ‘there exists . . . ’ type of claim since thepresentation of as many examples as possible does not guarantee that there is not atleast one case that corroborates this claim. Hence, a formal proof is usually applied.

Studies of the subject of proofs at school extensively discuss the difficultiesstudents encounter in the perception and comprehension of proofs [9–12]. An in-depth investigation of proving methods applied by high school students illustratesthey encounter difficulties, mainly in proving geometrical claims [13,14].Examination of students’ perceptions and beliefs related to the concept of proof aswell as to the validation and refutation methods indicates that only few of them areaware of the function of proof in mathematics and of valid arguments andexplanations [1,15–17]. Studies conducted by Chazan [18] and Fischbein [8]investigated how students prove and refute claims of the ‘for every . . . ’ and ‘thereexists . . . ’ types. They found that students manage to prove or refute claims byproviding only supporting- or counterexamples. However, when required to make aformal proof, they fail to do so. This is due to the fact that there is no reference to thesubject of proofs as a central subject by itself but rather as part of geometry studies.As mentioned, learners associate the concept of proof only with geometry and notwith additional topics in mathematics, such as algebra or differential calculus.Nevertheless, most studies related to ‘for every . . . ’ claims that were mathematicallyvalid and not to ‘there exists . . . ’ claims that were mathematically invalid and to theirrefuting methods by using counterexamples [19,20].

According to Fischbein [8], most students attest that checking individual casesrather than formal arguments convinced them that the claim was correct.In numerous cases, after having monitored a formal proof, the students needed tocheck also individual cases in order to be convinced of the certitude of the claim.Fischbein stipulates that the explanation resides in the way of thinking that is typicalto us as human beings. He maintains that the way of thinking, which characterizes usin our daily life, is inductive versus the deductive thinking that is typical ofmathematical proofs.

Zaslavsky and Ron [21] explored the way students use counterexamples in orderto refute a mathematical claim; how they understand the issue of counterexamples; inwhat way they form counterexamples; and what are the difficulties they encounter

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when refuting a claim by means of a counterexample. The researchers found thatnumerous students believe that checking a large number of cases (the term ‘large’ isdefined by the student and is entirely subjective), which corroborate the claim, provethat the claim is correct. They fail to understand that one counterexample issufficient for refuting the claim. In other words, most students think that if a specificclaim is valid in many cases, then it must be correct. Moreover, they fail tounderstand that if a specific claim is valid in specific cases (even if they are numerous)this does not guarantee that the claim is correct. Furthermore, Zaslavsky and Ron[21] found that in most cases students used intuitive and empirical arguments forproving all types of claims, namely, they relied mainly on examples and on checkingindividual cases. This is appropriate when dealing with claims the corroboration orrefutation of which can be performed with one supporting- or counterexample.Conversely, a claim whose corroboration or refutation required a deductive (general)proof, entailed many difficulties.

In his study of students’ methods of proving and refuting claims, Messinger [22]showed that some of the students were ‘over-deductive’. Accordingly, claims that canbe proven or refuted by one example ‘enjoy’ a more deductive proof by the students.Some of the students find it hard to accept that a proof by means of one example is agood proof.

Consequently, we should thoroughly check what is the students’ ability andknowledge to prove or refute geometrical claims and to what extent students find itdifficult to deal with claims that require a deductive proof.

This study aimed to examine whether high school students, learning geometry asa deductive subject, indeed know when a claim can be corroborated or refuted by aproof and when by an example. The study investigated which type of claim studentsuse, how they justify it and what type of thinking they manifest: inductive ordeductive.

Research questions

(1) How do high school students interpret the concept of proofs in geometry,that is, do they prove in a deductive way or present an example?

(2) Based on the answers, is it possible to identify students’ types of thinkingwhen providing proofs: inductive or deductive and even over-deductive?

3. Methodology

3.1. The research population

The research population consisted of twenty-five 10th-graders learning mathematicsat an intensive level. At the end of that grade they had to take the matriculation examat an advanced mathematics level (usually, students take the matriculation exam atthe end of the 11th or 12th grades). This was the beginning of the third year ofstudying Euclidean geometry. Eighty-eight percent of them had a score of over 85%in mathematics. During these 2 years, the students studied within the framework ofthe curriculum claims in geometry and the proofs thereof. They were exposed toproofs by negation (i.e., congruence of angles or sides) as well as to cases where, inorder to refute claims (theorems) in geometry, they had to present an example thatrefuted the claim. Throughout these 2 years they were required, as current exercising,to corroborate or refute claims using a counterexample by means of words,drawings, or formal proofs.

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3.2. Research tools

Two tools were used: a questionnaire and an individual structured, guided, and

focused interview.a. Questionnaire – consisted of four geometrical claims: two ‘for every . . . ’ claims

(Nos. 1 and 2), one ‘there exists . . . ’ claim (No. 3) and one combined, both ‘for

every . . . ’ and ‘there exists . . . ’ claim (No. 4). The three first claims were formulated

in pure geometric language, whereas the fourth claim was formulated from both

geometric and algebraic aspects. Moreover, the logical level of this claim was higher.

The students were asked to identify whether the claims were correct or incorrect and

to prove their answer. All the claims had been studied before during geometry

lessons and were not unfamiliar to the students. Below are the four claims presented

to the students as well as possible answers.

(1) ‘Every parallelogram is divided by the diagonal into two triangles of equal

area’ (Figure 1). This is a ‘for every . . . ’ claim and it is correct. The common

proof is by transferring altitudes to the opposite sides of the parallelogram

and proving that the areas are equal.(2) ‘Every two triangles of equal area are congruent’ (Figure 2). This is a ‘for

every . . . ’ claim and it is incorrect. This is an incorrect inverted theorem of the

theorem ‘two congruent triangles are of equal area’. In order to refute the

claim, the students could give an example of two triangles of equal area that

are non-congruent.(3) ‘There are two different parallelograms of equal area’ (Figure 3). This is a

‘there exists . . . ’ claim and it is correct. In order to justify its certitude,

students had to provide a supporting example of two different parallelograms

of equal area. For example, in order to corroborate this claim one can present

an example of a rectangular and a square of equal area, or a rhombus and

parallelograms that are equal in the length of one side and the altitude to

this side.(4) ‘For every three numbers a, b, c4 0 there is a triangle whose sides length is a,

b and c’. This is a ‘for every . . . ’ claim that is incorrect. In order to refute it,

the students could quote the theorem they had learnt, that is, ‘in every

triangle, the sum of every two sides is bigger than the third side’ or quote a

refuting example, explaining why it refutes the claim. For example, with the

length of the segments: 2, 3 and 10 we cannot create a triangle.

b. Individual structured, guided and focused interview – during which the

students were asked to explain their answers in the questionnaire regarding each of

the claims presented to them. The interview aimed to identify the students’ way of

thinking and the way they justified correct claims or refuted incorrect ones.

Figure 1. A parallelogram which is divided by diagonal into two triangles.

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The interviews were conducted with five students, chosen based on the way they hadchosen to prove or refute these four claims.

4. Findings

4.1. The ways of proving or refuting geometrical claims

The first research question dealt with the way high school students choose in order toconstruct a proof. Do they prove in a deductive way or present an example?

At the first stage, a questionnaire was used in order to find out to what extent thestudents identify correct and incorrect claims. The second stage examined how thestudents chose to justify their answer, regardless of the decision whether the claimwas correct or incorrect Table 1.

In claim No. 1 all the students identified that it was a correct claim. Most of themjustified their answer with a proof. However, it is important to point out that none ofthem, including those who did not provide a proof, gave an example

In claim No. 2 and claim No. 3 most of the students (more than 90%) identifiedwhether the claim was incorrect (claim No. 2) or correct (claim No. 3). They had tojustify their answer by giving an example rather than a proof. Indeed, most of themgave an example and not a proof and none of them gave both an example anda proof.

Below are some of the students’ answers written in the questionnaire to claim 2:‘In one triangle the altitude is 2 cm and the side is 3 cm, whereas in the second triangle,the side is 2 cm and the altitude is 3 cm. The triangles have the same area but theyare non-congruent’. Or: ‘One can create triangles that are non-congruent but haveequal area’.

Three students, whose answer was referred to as ‘other’, did not provide a proofand an example, they just repeated the claim in other words.

Some answers to claim 3: . . . . ‘Regarding parallelograms, if we take a parallelo-gram whose side is 8 cm and the altitude to this side is 5 cm, and another parallelogram

40mm

60mm

30mm

20mm

Figure 2. Two triangles and their areas.

80mm

40mm

40mm

20mm

Figure 3. Different parallelograms and their areas.

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whose side is 10 cm and the altitude to this side is 4 cm, we will get two parallelogramsof equal area but they are non-congruent’. Or: ‘In the case of parallelograms this is thesame as in triangles. One can create such parallelograms’.

Conversely, in claim No. 4 – length of a triangle sides – there is a differentiationbetween the students. When answering this claim, unlike in the first three ones, asmaller number of students answered that the claim was correct (60%) and fourstudents did not answer the claim at all. Seven students did in fact give a proof but asimilar number gave an example.

Unlike in the three previous claims, the students manifested some hesitation inchoosing their way of justification. Seven students gave both proof and example.

An example of a student’s answer correct to claim No. 4: ‘If there are threesegments that are 7, 4 and 11 cm, respectively, it is impossible to create a triangle withthem. They cannot be connected . . . ’

Also, in claim No. 4 (triangle sides), only 15 students answered that the claim wasincorrect. However, although they thought that the claim was incorrect, they foundthat a proof was needed and that an example was insufficient. Most of therespondents justified their answer by means of an example, as one of the studentswrote: ‘There is a triangle whose sides are 3, 4 and 5 cm and that is even a right-angledtriangle . . . ’ or ‘an isosceles triangle whose sides are 7, 7 and 2 cm. . . .’

Most of the respondents who believed that claim 4 was incorrect justified theiranswer by giving an example, or both an example and a proof (13 students, six whogave just an example and seven who gave both an example and a proof). It isimportant to note that all those who proved this claim (total of nine students, out ofthem two who gave only a proof and seven that gave both a proof and an example)

Table 1. Distribution of answers (proof, an example, etc.).

Presentationof

Answer toclaim No.

Correct

Did notanswerProof Example

Bothproof andexample

Other justansweredwithoutproof orexample

1. Parallelogramdiagonals (correctclaim)

Correctanswer

21 (84%) – – 4 (16%) –

Incorrectanswer

– – – –

2. Triangles of equalarea (incorrectclaim)

Correctanswer

1 (4%) 21 (84%) – 1 (4%) –

Incorrectanswer

– – 2 (8%)

3. Parallelogram ofequal area (correctclaim)

Correctanswer

1 (4%) 21 (84%) – 1 (4%) –

Incorrectanswer

– 2 (8%) – –

4. Triangle sides(incorrect claim)

Correctanswer

2 (8%) 6 (24%) 7 (28%) – 4 (16%)

Incorrectanswer

5 (20%) 1 (4%) – –

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gave a correct answer. This approach is entirely different than the approach adoptedin the first three claims, whereby students preferred to give a proof or an example butnone of them gave both. On the other hand, five students, who thought that claimNo. 4 was correct, preferred to justify their answer by means of a proof that wasobviously wrong.

‘We will prove by showing that in a triangle, opposite a wide angle there is a longside. If angle A is bigger than angle B, then a4 b and, in the same way, if angle B isbigger than angle C, then b4 c’. This pattern is entirely different than the pattern ofstudents who answered, rightly, that the claim was incorrect.

Moreover, the findings indicate that the answer patterns in the first three claimswere different than those in the fourth claim. In the case of the first three claims,most of the students chose the right ways for justifying their answer, whether theclaim was correct or not, avoiding the presentation of arguments. On the other hand,in the fourth claim, even those students who believed that the claim was incorrect,used both proof and example for justifying their answer; some of them even tendedto provide an argument for it. In addition, the students encountered some difficultywith this claim: both regarding the decision whether it was correct or not and the wayof justifying their answer.

‘This is simple because there is a theorem that says it . . . then we can simply explainthe theorem’ . . .

To sum up, analysis of the interviews illustrates that in those cases where theclaim was simple and familiar to the students, they applied one way of proving.However, when the claim was difficult or complicated, they used several ways forpresenting the proof.

4.2. The thinking pattern

The second research question checked whether it is possible to identify, based on thefive students’ answers in the questionnaire and the interview, their profile of thinkingpatterns. The interviews supported the characterization of the interviewees’ thinkingpatterns (A, B, C, D, E).

Table 2 illustrates the ways that the five students chose in order to prove or torefute the four claims, as well as the students’ categorization on the basis of theiranswer to the claims (deductive/over-deductive/mixes). Four parameters wereexamined in every answer.

(1) The student was right/wrong in determining whether the claim was correct orincorrect).

(2) The student gave/did not give a deductive proof in order to justify his or herdetermination.

(3) The student gave/did not give an example in order to justify his or herdetermination.

(4) The student gave/did not give an explanation of the example.

Student A: Analysis of student A interview shows that this student knew toanswer accurately whether the claim was correct or incorrect. Moreover, he knewwhen to use a proof and when an example in order to corroborate or refute a claim.Nevertheless, in his answers during the interview he indicated that he preferred toexplain by means of letters rather than numbers when presenting examples.

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Table

2.Thethinkingpatternsofthefivestudents.

Student

Profile

of

thinking

patterns

Claim

no.1

Claim

no.2

Claim

no.3

Claim

no.4

AOver-deductive

Correctansw

erCorrectansw

erCorrectansw

erCorrectansw

erCorrectproof

Gaveageneralexampleand

explained

itGaveanexample

without

anyexplanation

Gaveageneralexample

and

anexplanation

BDeductive

Correctansw

erCorrectansw

erCorrectansw

erCorrectansw

erGaveanexplanation

Gaveanexample

and

explained

itGaveanexample

without

explanation

Gaveaproofandan

example

CDeductive

Correctansw

erCorrectansw

erCorrectansw

erDidn’tansw

erIncorrectproof

Gaveanexample

without

explanation

Gaveanincorrectexample

withoutexplanation

DMixed

Correctansw

erCorrectansw

erCorrectansw

erIncorrectansw

erCorrectproof

Gaveanexample

without

explanation

Gaveanexample

without

explanation

Made2drawingsas

examples

EOver-deductive

Correctansw

erCorrectansw

ergavea

generalexplanation

Correctansw

erCorrectansw

erCorrectproof

Gaveanexplanation

Usedthetheorem

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In his opinion, a proof is more acceptable because ‘it is more general’ whereas ‘I thinkthat an example is something very specific while a proof is more general, more serious.That is how I feel, I don’t really know to explain why’. When giving examples, hepreferred to use letters – parameters – and not numerical examples because it seemedmore ‘serious’, ‘more appropriate’. According to him, there are cases where ‘you haveto emphasize in order to be convinced and to give an example or general explanation inorder to show that it is always like that’. Furthermore, the interview illustrated that heperceived the examples with letters as some kind of proof: ‘A general example is somekind of proof, I showed it in a general way, I generalized’. In the questionnaire heanswered claim No. 4 by giving a numerical example of segments: 5, 4 and 10 cm,which cannot form a triangle. He also gave a general explanation. He explained hethought that ‘the difference between the ‘long’ side and each of the sides should be verybig’. During the interview he also recalled the theorem: ‘ha . . . sure, there is a theoremthat says the sum of two sides of a triangle should be bigger than the third side and thisis actually what I have tried to show in a general way. It’s like a proof, isn’t it?’ He wasasked if making a proof was important to him. His answer was: ‘I think it ispreferable, if we want to convince’. When asked if an example like the one he hadgiven was insufficient or maybe it was preferable to give a general explanation, heresponded: ‘A general explanation is like a proof. However, in this particular case itwas preferable to give both in order to show that it is always like that.’

To sum up, this student gave an ‘over-deductive’ answer, relating to those caseswhereby students are trying to deductively prove every claim. Otherwise, they are notconvinced whether the claim is correct or incorrect.

Student B: Analysis of student B interview shows that this student knew to answerwhen a claim was correct or incorrect. Moreover, he knew when to use a proof andwhen an example in order to corroborate or refute a claim. Nevertheless, at the endof the interview he pointed out: ‘The first claim was general, ‘‘for every . . . ’’ type, andothers were not. Regarding the triangles that have equal area but are non-congruent, anexample would suffice’. He expressed his disappointment with claim No. 4: ‘It wasboring’ because it was enough to quote the theorem dealing with the length of atriangle sides. It is important to emphasize that he wrote the examples to claim No. 2and claim No. 3 in a deductive proof format. This style of writing highlights oncemore that geometry is related to proofs and, therefore, geometrical arguments arepresented in a different format than algebraic ones. It is significant to say that, unlikethe deductive thinking pattern manifested in the style of writing the example, hefailed to provide explanations or arguments for his examples: non-congruentparallelograms as well as non-congruent triangles. During the interview he could notexplain the example, although he indicated: ‘I know that we have to explain but I don’tknow how . . .’

To sum up, this student represents the ‘deductive student’, who knows when tomake a deductive proof or when a numerical example is sufficient for proving orrefuting a claim.

Student C: The interview with student C, who gave unsupported proofs, shows thathe did not think thoroughly about his answers. The interview strengthened thefeeling that the student knew whether a claim was correct or incorrect. Moreover, heknew when to use a proof and when an example was sufficient in order tocorroborate or refute a claim. He did not refer to claim No. 4. When asked to explainhis examples, expressing some doubt whether his examples were appropriate,

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his reaction was: ‘Why? It’s not sufficiently clear?’ To the question how he knew inthe example of the triangles example of equal area why they were non-congruent, hesaid: ‘I built it like that,’ or ‘it’s really annoying.’ During the interview he was askedwhy he failed to answer claim No. 4 and he replied: ‘It was formulated in acomplicated way. Every 3 numbers and . . . I thought it was a confusing claim and didnot feel like thinking’. When asked to answer the claim in the interview, he thoughtsome time and immediately associated and quoted the appropriate theorem. Thisstudent was in fact characterized as ‘deductive’. Nevertheless, it was obvious that hewas too lazy to think in-depth, preferring to engage in clear and obvious issues.According to him, he prefers algebra to geometry because, ‘Geometry is veryexhausting. We have to explain or give arguments for everything. Even if everythingseems logical – then you either did not think of something or you were wrong’. Heclearly distinguishes between geometry and algebra mainly from the thinking aspectand the need to be very specific in geometry, something that he believes algebra doesnot require.

Student D: pointed out in her interview that she did not like geometry because‘Geometry is harder and does not interest me very much’. The student was right in thefirst three claims. She knew that in order to refute claim No. 2 and prove claim No. 3an example was sufficient whereas in order to prove claim No. 1, a proof was needed.When asked during the interview why she did not explain the examples for claimsNos. 2 and 3, she said that, ‘I thought it was obvious that I drew altitudes down andshowed that if the dimensions were as indicated, the triangles are of equal area,I thought it was obvious . . . it’s easy to see’. She pointed that, ‘It’s annoying’. From herpoint of view it was clear and, in this case, her intuition was right. When asked whyin claim No. 1 she did not give an example, she specified: ‘What? Why there is noexample? Because we should say that in every parallelogram, the diagonal divides it intotwo triangles of equal area, so obviously one example does not illustrate it’. Regardingclaim No. 4, also in the interview she experienced a cognitive conflict, got confusedand did not know whether a proof or an example was needed. She tried to identify arule when to prove and when to exemplify. She said that the claim was confusing:‘They said positive numbers, so why is there a problem? I need to show many triangles,I can draw triangles. I drew and tried to make is general, as one does in geometry.I don’t know, I just guess, so a proof is needed, especially if it is correct’. In fact,without being explicit, she tried generalizing: if the claim is correct, then one needs toprove and if it is incorrect, then one needs an example. When the interviewerpresented to her the case where the claim was correct and yet she used an example,she gave up but immediately found an example for refuting claim No. 4. The exampleshe gave led her to the theorem dealing with triangle sides. She quoted it and summedup the discussion by saying, ‘I told you that I don’t like geometry’. This student wascharacterized as ‘mixed’. Her interview illustrated that, although at the beginning herintuition worked accurately, it was easy to confuse her. She was unable to decidewhen to give a proof and when to settle for one example in order to corroborate orrefute a claim. ‘You managed to confuse me. Then, can I or cannot I?’.

Student E: Analysis of student E interview shows that this student knew when aclaim was correct or incorrect. Moreover, he knew that in claim No. 1 a proof wasnecessary in order to justify the claim. In the other claims he did not prove but gave ageneral example in letters rather in numbers. During his interview he repeated theexplanations given in the questionnaire. When asked whether his explanations were

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sufficient, he responded: ‘Yes, because it is general. In fact I tried to illustratesomething here. Perhaps one can call it an example? Some sort of general example’.To the question if it is possible to convince that a claim is correct by means of anumerical example, he replied: ‘Perhaps to convince. However, in geometry, one has toprove in a general manner . . . because this is the way it is done in geometry’. Regardingclaim No. 4, he indicated: ‘It is simple. There is a theorem dealing with this. So I justhad to explain the theorem’. This student is clearly ‘over-deductive’, wanting to proveeverything in a general manner. Hence, in those cases where he failed to make adeductive proof, he settled for general explanations and avoided giving a specificexample.

To sum up, based on the interviews, one can clearly identify the three studenttypes according to their chosen ways of proving or refuting: ‘deductive’, ‘over-deductive’ and ‘mixed’. Out of the five interviewed students, two were characterizedas ‘deductive’, two as ‘over-deductive’ and one as ‘mixed’.

The students characterized as ‘deductive’ knew when to use a proof and when touse an example; however, they failed to explain the reason for it. The intuition ofthese students was accurate. The students characterized as ‘over-deductive’ tended touse proofs and general examples, even when a single numerical example wassufficient. Students with this thinking pattern found it hard to accept an example as aproof and they defined this way as ‘not serious’. The third group of ‘mixed’ studentscomprised those who were inconsistent in choosing the way for proving or refuting aclaim.

5. Discussion and conclusions

The main question in the present study was whether students learning geometry as adeductive subject, know when a claim can be corroborated or refuted by proof andwhen by example. In order to explore this question, questionnaires were adminis-tered and interviews were conducted while relating to sub-questions. The question-naires given to the students examined how they justify or refute geometrical claims.The interview conducted with some of them engaged mainly in the reasons forchoosing the way of proving or refuting the claim. Moreover, the study investigatedthe assumption that students manifest ‘over deduction’, according to which they tendto prove every claim, even if one example is sufficient for corroborating it. It alsochecked whether students who provide an example settle just for an example or, inaddition, explain its certitude and its relation to the claim.

Analysis of the questionnaire illustrates that in the case of ‘easy’ claims,formulated in a familiar and clear way, the students knew when to use a deductiveproof or an example and did not apply both options (proof and example). The givenproofs were well structured and the students demonstrated confidence in their way.However, some students slightly hesitated when writing an example (manydeletions).

The findings of the present study are not in line with the study conducted byZaslavsky and Ron [21], showing that most of the students fail to understand thatone counterexample is sufficient for refuting a claim. The difference might be due tothe fact that the research population in the present study comprised high-achievingstudents, learning on an advanced level and considered as students with aparticularly good mathematical competence. It is interesting to note that only one

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student could explain during the interview his decision to use an example or proof.The other students could not explain it but during the interview they clearly showedthat they understood the difference between the two types of argument.

Different findings were shown in relation to claim No. 4. The students somewhathesitated in choosing the way of justification. Some of them preferred not to answersince they could not find counterexamples nor did they manage to prove. Most of thestudents who proved or quoted the appropriate theorem, also gave an example. Thisindicates that the students were not sure of the certitude of the claim if they did nothave a specific example. Consequently, they tried to ‘mobilize additional means ofconvincing’. This was defined by Fischbein and Kedem [1] as formal and inconsistentthinking pattern. As to ‘over-deductive’ pattern, one of the assumptions was thatstudents associate deductive proof as the only way of convincing in geometry. Anexample in letters – a general one – is for them a ‘proof’ rather than an example.

To sum up, only in the first claim, which was clear and familiar, the studentsdemonstrated a consistent formal thinking pattern and did not need a supportingexample. Moreover, the present study found that the harder the claim, the more‘channels’ of explaining and convincing the students use, rather than settle for aformal proof or one example. Conversely, in claims that they deemed insignificant,the students found it unnecessary to give a formal explanation and did not think verythoroughly. These findings are in line with the study of Fischbein [8], who showedthat checking individual cases convinces the students of the certitude of the claimmore than formal arguments. Consequently, when the claim is unfamiliar or isdifferently formulated, students feel more confident when providing also an example.

These findings raise the question whether this conception of the notion of prooforiginates in the curriculum and, if it does, how can it be changed. Perhaps the senseof needing a formal proof can be developed if students find out phenomena bythemselves, present assumptions and enhance the need for generalization, leading toa formal proof. Perhaps in this way students will acquire tools for deciding when touse a proof and when to use an example in order to validate or refute claims.Another point worthy of thought is associating the concept of proof only with thediscipline of geometry. If students learn algebraic formulas and techniques by way ofdeductive proofs, then proof might turn into a means and not the end itself. Thus,students will learn to identify the need for giving a proof or an example.

6. Recommendations and limitations

In addition to the above conclusions, there are, however, some reservationsregarding the findings, taking into consideration several factors: the small number ofparticipants and the fact that the study was conducted with high-achieving students.The questionnaire engaged only in geometry. Consequently, it is recommendedconducting a similar study with a larger student group. Furthermore, another studycombining claims in algebra and geometry should also be conducted, investigatingwhether the students use different ways for proving the claims.

References

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in Proceeding of the 6th Conference of the International Group for the Psychology ofMathematics Education, A. Vemandel, ed., Universitaire Instelling Antwerpen, Antwerp,

Belgium, 1982, pp. 128–131.

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