Educational Studies in Mathematicsmath.fau.edu/yiu/PSRM2015/yiu/New Folder (4)/Downloaded Papers... · Proof is an important part of mathematics itself, of course, and so we must

Educational Studies inMathematics

Volume 44, Numbers 1 and 2

March 2002

Special Issues on

Proof in Dynamic geometryEnvironments

Guest editors:

Keith Jones,

Angel Gutierrez,

Maria Alessandra Mariotti.

Table of Contents

Keith Jones, Angel Gutirrez, Maria Alessandra Mariotti,Guest Editorial, 1 - 3

Gila Hanna,

Proof, Explanation and Exploration: An Overview, 5 - 23

Maria Alessandra Mariotti,

Introduction to Proof: The Mediation of a Dynamic SoftwareEnvironment, 25 - 53

Keith Jones,

Providing a Foundation for Deductive Reasoning: Students’Interpretations when Using Dynamic Geometry Softwareand Their Evolving Mathematical Explanations, 55 - 85

Ramn Marrades and ngel Gutirrez,

Proofs produced by secondary school students learning geometryin a dynamic computer environment, 87 - 125

Nurit Hadas, Rina Hershkowitz, Baruch B. Schwarz,

The role of contradiction and uncertainty in promoting the need toprove in Dynamic Geometry environments, 127 - 150

Colette Laborde,Dynamic Geometry Environments as a Source of Rich Learning

Contexts for the Complex Activity of Proving, 151 - 161

GUEST EDITORIAL

Since its creation in 1968, Educational Studies in Mathematics (ESM) hasbeen a leading journal in the field of Mathematics Education. In the pagesof ESM the results of new and important research in Mathematics Edu-cation have been reported and the most relevant research questions raisedand discussed. Somewhat in parallel, the International Group for the Psy-chology of Mathematics Education (PME) has become, since its inauguralmeeting in 1977, one of the main conference forums for researchers inMathematics Education. Every year at PME, researchers from all over theworld discuss the latest research questions and results, and define futureresearch directions. As a result of joint effort by ESM and PME, ESMhas initiated a programme of periodically publishing PME special issuesaimed at showcasing important and substantial aspects of topics workedout by the PME community.

As editors of this PME special issue, we are very pleased to presentthe first outcome of the PME Special Issue series. This issue is devoted toanalysing the influence of dynamic geometry software (DGS) on students’conceptions of mathematical proof while the students are solving geometryproblems involving proofs in an environment mediated by such software.In particular, this Special Issue gathers together, extends and comparesa range of recent research, much of it presented at PME conferences orbenefiting from discussions in that forum.

The paper by Gila Hanna serves as an introductory paper and dealswith some theoretical aspects of questions considered in the four followingresearch papers. Hanna first addresses the question of the role of proof insecondary school mathematics, and the contrast between abstract proofsand heuristics, explorations, and visual proofs. Her remarks on epistemo-logy towards the end of the paper constitute an important element of thebackground to this Special Issue.

The four research papers explore the central question of whether theopportunities offered by DGS environments to ‘see’ mathematical proper-ties so easily might reduce or even replace any need for proof or, on thecontrary, whether such a facility might open up new ways of meaningfulapproaches to promoting students’ understanding of the need for and the

Educational Studies in Mathematics 44: 1–3, 2000.© 2001 Kluwer Academic Publishers. Printed in the Netherlands.

2 EDITORIAL

roles of proof. Each of the papers addresses this issue within a distincttheoretical framework.

In the first research paper, and employing a Vygotskian perspective,Maria Alessandra Mariotti presents an analysis of the relevant mediationof several components and commands of the DGS on the interactions ingroups of students. She highlights correspondences amongst the DGS menucommands used by students and axioms and theorems they subsequentlyuse in their justifications.

Keith Jones presents an environment where students are faced withthe question of the classification of quadrilaterals by means of a set ofproblems of increasing difficulty. The analysis of students’ answers showsthat using dynamic geometry software helps students to progress in theirunderstanding of the dependence relationships among components of a fig-ure and amongst families of figures, and so advance towards a progressiveabstraction in their justifications.

Ramón Marrades and Ángel Gutiérrez report on a teaching experimentdesigned to enable students to produce deductive justifications of the cor-rectness of their constructions. An analytic framework, which integratesand expands previous frameworks, is used to analyse student answers andto show that the way in which they use the DGS determines their solutionsand justifications, and how the quality of students’ justifications improvesover time.

Nurit Hadas, Rina Hershkowitz and Baruch Schwarz present an ap-proach to enabling students to produce deductive justifications by present-ing them with problems that reveal surprising, contradictory or uncertainresults. The authors present different categories of students’ answers to thiskind of problems when solved in a DGS environment and show how theDGS allows students to move between certainty and uncertainty, betweenconjecture and checking the conjecture.

In the final paper in this Special Issue, Colette Laborde, by offering aglobal integrating overview of the four research papers, describes to whatextent they complement each other. Using the papers, Laborde illustrateshow it is possible to build many different DGS teaching environments,adapted to specific necessities of students, where students can gain a betterunderstanding of the deductive structure of mathematics, and the need forjustifications/proofs in mathematics. Furthermore, Laborde highlights theusefulness of DGS in breaking down the traditional separation betweenaction (as manipulation associated to observation and description) anddeduction (as intellectual activity detached from specific objects).

While the individual papers in this special issue are the results of theirauthors’ own research activity, the quality of the work has been enhanced

EDITORIAL 3

by the environment of the PME annual conferences where many valuablediscussions have taken place over a number of years. In particular, theeditors of this Special Issue would like to record their appreciation of manyPME colleagues, particularly Paolo Boero, Celia Hoyles, John Pegg andMichael de Villiers, who contributed to the development of their ideas.

Finally, as guest editors we would especially like to thank TommyDreyfus who, as collaborating ESM editor for this first PME Special Issue,worked closely with us, advising us on procedures, and sharing much ofthe decision-making. His expertise was a great help to us during the timeof preparation of this special issue.

This Special Issue provides a range of evidence that working with dy-namic geometry software affords students possibilities of access to theor-etical mathematics, something that can be particularly elusive with otherpedagogical tools. Yet it has to be noted, as Hanna points out in her in-troductory paper, that the examples of successful access to mathematicaltheory presented in the four research studies did not happen without care-fully designed tasks, professional teacher input, and opportunities for stu-dents to conjecture, to make mistakes, to reflect, to interpret relationshipsamong objects, and to offer tentative mathematical explanations. The re-search presented in this Special Issue needs replication and amplification.In particular, research in the use dynamic geometry software to supportthe development of students’ mathematical thinking could usefully focuson the nature of the tasks students tackle, the form of teacher input and therole of the classroom environment and culture. For teachers in particular,that something works is one thing – further examples of how it can bemade to work in the variety of classrooms are crucial.

KEITH JONES, ÁNGEL GUTIÉRREZ AND

MARIA ALESSANDRA MARIOTTI

GILA HANNA

PROOF, EXPLANATION AND EXPLORATION: AN OVERVIEW

ABSTRACT. This paper explores the role of proof in mathematics education and providesjustification for its importance in the curriculum. It also discusses three applications ofdynamic geometry software – heuristics, exploration and visualization – as valuable toolsin the teaching of proof and as potential challenges to the importance of proof. Finally,it introduces the four papers in this issue that present empirical research on the use ofdynamic geometry software.

KEY WORDS: dynamic geometry, proof, technology, visualization

INTRODUCTION

There has been a recent upsurge in papers on the teaching and learningof proof: Between 1990 and 1999 the leading journals of mathematicseducation published over one hundred research papers on this topic. Inaddition, the International Newsletter on the Teaching and Learning ofMathematical Proof, which Nicolas Balacheff has maintained as a Web sitesince 1997, has been visited over five thousand times to date. This newslet-ter publishes information on theoretical and empirical research on proof,primarily on papers and books, and is updated six times a year. Theseare certainly indications that proof is a prominent issue in mathematicseducation.

Some of this activity has to do with the very ‘raison d’être’ of proof.This is not surprising, since certain developments in both mathematics andmathematics education have called into question the role of proof. Let mestart, then, by stating categorically that proof is alive and well in mathem-atical practice, and that it continues to deserve a prominent place in themathematics curriculum. One of our key tasks as mathematics educators,however, is to understand the role of proof in teaching, so that we canenhance its use in the classroom.

Proof is an important part of mathematics itself, of course, and so wemust discuss with our students the function of proof in mathematics, point-ing out both its importance and its limitations. But in the classroom the keyrole of proof is the promotion of mathematical understanding, and thus our


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most important challenge is to find more effective ways of using proof forthis purpose.

One of these potentially more effective ways is to use dynamic geo-metry software, which opens up entirely new approaches to the teaching ofproof. The four research papers in this special issue address this opportun-ity, discussing empirical research into the degree to which such softwarecan be used effectively with students to help them develop sound math-ematical reasoning, produce valid proofs of geometric propositions, andenhance their understanding of mathematics in the broadest sense. To placethis issue in context, these articles also review the research literature onthe range of difficulties encountered when teaching proof in the secondaryclassroom.

My paper, on the other hand, is not empirical in focus. It deals, rather,with some theoretical aspects of questions considered in the four researchpapers. It first explores the role of proof in mathematics education, andprovides some justification for the importance of proof in the curriculumand in particular for its usefulness in promoting understanding. It then goeson to look at heuristics, exploration and visualization, three important po-tential applications of dynamic software (though of course not exclusive toit), discussing each of them both as a valuable tool in the teaching of proofand as a potential challenge to the importance of proof in the mathematicscurriculum. The paper continues with some observations on the need fora clear view, on the part of educators, of the nature of mathematics ingeneral and of the relationship between deduction and experimentationin particular, and concludes with a brief review of the empirical researchpresented in detail in the four research papers that follow.

THE ROLE OF PROOF

Over a number of years I had been asking myself what role proof oughtto play in mathematics education. Of course this raised the underlyingquestion of the role of proof in mathematics itself. An examination of thephilosophy and history of mathematics made it clear to me, first of all,that there long have been and still are conflicting opinions on the role ofproof in mathematics and in particular on what makes a proof acceptable.This topic is discussed at some length in Rigorous proof in mathematicseducation (Hanna, 1983), which was in essence a critique of the view ofproof adopted by the ‘new math’ movement of the 1950s and 1960s, andin particular of its emphasis on rigour. Then, through a closer examinationof mathematical practice, I came to the further conclusion that even in theeyes of practising mathematicians rigorous proof, however it is defined,

PROOF, EXPLANATION AND EXPLORATION 7

is secondary in importance to understanding. It became clear to me that aproof, valid as it might be in terms of formal derivation, actually becomesboth convincing and legitimate to a mathematician only when it leads toreal mathematical understanding.

Mathematical understanding

All four research papers in this issue explore the effect on mathematicalunderstanding of the use of dynamic geometry software in the classroom.Mathematics teachers are well aware that the term ‘mathematical under-standing’ is somewhat elusive. This is not surprising, since the nature ofunderstanding is a topic of discussion among practising mathematiciansas well. Mathematicians, however, know that there is such a thing, andin fact most share the view that a proof is most valuable when it leadsto understanding, helping them think more clearly and effectively aboutmathematics (Rav, 1999; Manin, 1992, 1998; Thurston, 1994).

Mathematicians, then, see proofs not only as syntactic derivations (se-quences of sentences, each of which is either an axiom or the immediateconsequence of preceding sentences by application of rules of inference).Rather, they see proofs as primarily conceptual, with the specific technicalapproach being secondary. Proofs are the “mathematician’s way to dis-play the mathematical machinery for solving problems and to justify that aproposed solution to a problem is indeed a solution” (Rav, 1999, p. 13).

Rav suggests we think of proofs as “a network of roads in a publictransportation system, and regard statements of theorems as bus stops”.A similar metaphor is used by Manin (1992) when he says that “Axioms,definitions and theorems are spots in a mathscape, local attractions andcrossroads. Proofs are the roads themselves, the paths and highways. Everyitinerary has its own sightseeing qualities, which may be more importantthan the fact that it leads from A to B.”

These metaphors speak directly to mathematics education, where a proofis important precisely for its ‘sightseeing’ qualities. Clearly students oughtto be taught the nature and standards of deductive reasoning, so that theycan tell when a result has or has not been established. But proof can makeits greatest contribution in the classroom only when the teacher is able touse proofs that convey understanding.

The functions of proof

It is useful, when attempting to set out the role of proof in the classroomin a systematic fashion, to consider the whole range of functions whichproof performs in mathematical practice. Proof in the classroom wouldbe expected to reflect all of them in some way. But these functions are

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not all relevant to learning mathematics in the same degree, so of coursethey should not be given the same weight in instruction (de Villiers, 1990;Hersh, 1993).

As mentioned above, mathematicians clearly expect more of a proofthan justification. As Manin (1977) pointed out, they would also like itto make them wiser. This means that the best proof is one that also helpsunderstand the meaning of the theorem being proved: to see not only thatit is true, but also why it is true. Of course such a proof is also moreconvincing and more likely to lead to further discoveries. A proof mayhave other valuable benefits as well. It may demonstrate the need for betterdefinitions, or yield a useful algorithm. It may even make a contribution tothe systematization or communication of results, or to the formalization ofa body of mathematical knowledge.

The following is a useful list of the functions of proof and proving (Bell,1976; de Villiers, 1990, 1999; Hanna and Jahnke, 1996):

• verification (concerned with the truth of a statement)• explanation (providing insight into why it is true)• systematisation (the organisation of various results into a deductive

system ofaxioms, major concepts and theorems)• discovery (the discovery or invention of new results)• communication (the transmission of mathematical knowledge)• construction of an empirical theory• exploration of the meaning of a definition or the consequences of an

assumption• incorporation of a well-known fact into a new framework and thus

viewing it from a fresh perspective

But just as such a richly differentiated view of proof and proving couldarise only as the product of a long historical development, so must everystudent just entering the world of mathematics start with the fundamentalfunctions: verification and explanation. (Scholars pointed out the import-ance of clarification, as distinct from justification, as early as the 17thcentury.) But in the classroom, the fundamental question that proof mustaddress is surely ‘why?’. In the educational domain, then, it is only naturalto view proof first and foremost as explanation, and in consequence tovalue most highly those proofs which best help to explain.

Some proofs are by their nature more explanatory than others. An in-sight into what distinguishes an explanatory proof is provided by Steiner(1978), who says that such a proof will make “. . . reference to a char-acterizing property of an entity or structure mentioned in the theorem,such that from the proof it is evident that the results depend on the prop-erty” (p. 143). Closely related to Steiner’s definition is the concept of


‘inhaltlich-anschaulicher Beweis’. Wittmann and Müller (1990) use thisterm to characterize a proof in which “[the] method of demonstration callsupon the meaning of the term employed, as distinct from abstract methods,which escape from the interpretation of the terms and employ only theabstract relations between them”.

An example is the usual proof that the three angle bisectors of a trianglemeet at a single point, the incentre of the triangle. This proof, which makesuse of the characterizing property that an angle bisector is the locus of allpoints equidistant from the edges of the angle, has been found to be bothconvincing and illuminating, helping students see why the theorem mustbe true.

For teachers there are great advantages to be gained by choosing ex-planatory proofs, as Hanna (1990) has discussed in some detail. Of courseone cannot always find an explanatory proof for every theorem one wishesto present. In many mathematical subjects some theorems need to be provedusing contradiction, mathematical induction or other non-explanatory meth-ods. It so happens that geometry enjoys a special position in this regard,however, in that most of its proofs are explanatory.

HEURISTICS VS. PROOF

Some educators, concerned for the effective allocation of valuable class-room time, maintain that in the competition for classroom focus the topicof proof should take second place to heuristics. They believe that they haveto make a choice between developing investigative and problem-solvingskills (which in their opinion make mathematics look ‘useful’, ‘enjoyable’,and more of ‘a human activity’), and instilling the ostensibly less usefuland enjoyable skills needed to construct proofs (Simon and Blume, 1996;Simpson, 1995). They would seem to see proof as a chore, and as animpediment to understanding rather than as a route to it.

Simpson (1995) differentiates between ‘proof through logic’, whichemphasizes the formal, and ‘proof through reasoning’, which involves in-vestigations. The former is ‘alien’ to students, in his view, since it has noconnection with their existing mental structure, and so can be masteredonly by a minority. He believes that the latter appeals to the ‘natural’learner, however, because it embodies heuristic argument, and so is ac-cessible to a greater proportion of students.

Other educators, in expressing the view that deductive proof need nolonger be taught, have focussed not only on reasoning, but also on justifi-cation. Their belief is that heuristic techniques are more useful than proofin developing skills even in justification, where proof might have been

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expected to have enjoyed the advantage. Their argument is that much ofwhat parades in the classroom as the teaching of proof is actually the rotelearning of mathematical proofs, devoid of any educational value. Theysee a more significant educational role for investigation, exploration andinformal justification, all of which make use of intuition and yet are morelikely than proof, in their view, to engender mathematical insight and eventechnical fluency. Accordingly they would support cultivating a perceptionof mathematics as a science that stresses heuristics and the inductive ap-proach. This view has found expression in the NCTM Standards (1989)and the British National Curriculum (Noss, 1994).

The NCTM Standards

By the time the Standards (1989) was published by the National Councilof Teachers of Mathematics (NCTM) in the United States, the conceptof proof had all but disappeared from the curriculum (Greeno, 1994) orshrunk to a meaningless ritual (Wu, 1996). The NCTM did not seek toreverse this situation in the curriculum as a whole. It even proposed ashift in the teaching of geometry, the traditional stronghold of proof in theUnited States, recommending that less emphasis be given to two-columnproofs and to Euclidean geometry as an axiomatic system.

On the other hand, the Standards did propose greater emphasis on thetesting of conjectures, the formulation of counterexamples and the con-struction and examination of valid arguments, as well as on the ability touse these techniques in the context of non-routine problem solving. Thereare even two topics, among the seven recommended for greater attention,which have a distinct flavour of proof: (1) short sequences of theorems,and (2) deductive arguments expressed orally and in sentence form (pp.126–127).

But the NCTM’s strategy for the reform of the mathematics curriculumstopped short of mathematical proof as such. Its approach was to stress mo-tivation and ‘heuristic argument’. As a result the Standards (1989) failedto exploit the potential of proof as a teaching tool. Nor did this documentreflect mathematical practice, where heuristic arguments, important as theyare in discovery and understanding, are no substitute for proof.

The new version of the NCTM Principles and Standards (2000) hasremedied this situation by recommending that reasoning and proof bea part of the mathematics curriculum at all levels from prekindergartenthrough grade 12. The section of this document called ‘Reasoning andProof’ states that students should be able to:

• recognize reasoning and proof as fundamental aspects of mathemat-ics;


• make and investigate mathematical conjectures;• develop and evaluate mathematical arguments and proofs;• select and use various types of reasoning and methods of proof.

British National Curriculum

Proof has lost ground to heuristics in the United Kingdom as well. Noss(1994) observed that “some educators are convinced that proof in the cur-riculum is a barrier to investigative and creative activity, and that it isinherently opposed to the spirit of exploration and investigation which haspermeated the UK mathematics curriculum for much of the recent past”(p. 6).

The London Mathematical Society, the Institute of Mathematics and itsApplications and the Royal Statistical Society reacted to this tendency in ajoint paper published in October, 1995. Addressing the British mathemat-ics curriculum from primary school through university, the paper expressedconcern for what it termed “a changed perception of what mathematicsis – in particular of the essential place within it of precision and proof”(quoted in Barnard et al., 1996, p. 6), and also criticised schools for failingto prepare students adequately for university mathematics.

This statement from the three professional bodies prompted the edit-ors of Mathematics Teaching to invite a number of people to discuss theissue of proof in mathematics and to publish their remarks under the titleTeaching Proof (Barnard et al., 1996). The extreme viewpoint was perhapsthat of MacKernan, who deplored “proofs of the ghastliness required bytoday’s academic journals”, and went so far as to ask, “So, do we reallyneed proof at all? Especially in schools? . . . Why on earth can’t we – theoverwhelming majority – simply be allowed to accept that something isintuitive, or very probably true, or just simply obvious?” (p. 16).

The views of mathematicians: A recent debate

It is instructive to examine what some mathematicians have had to sayabout the relative importance of heuristics and proof in mathematics. Jaffeand Quinn (1993), while acknowledging that the use of rigorous proofhad been a blessing to mathematics, bringing “a clarity and reliability un-matched by other sciences”, identified a trend “toward basing mathematicson intuitive reasoning without proof” (p. 1). They attributed this trend tothe influence of the less rigorous standards of reasoning commonly em-ployed in the physical sciences. In reaction to it, they suggested that twodistinct types of mathematical endeavour, employing different types of jus-tification, be accorded legitimacy, but that they be clearly distinguished. In

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their proposal, mathematical results based upon speculative and intuitivereasoning or upon the examination of test cases would be referred to as‘theoretical mathematics’, while the label ‘rigorous mathematics’ wouldbe applied to the results of what has traditionally been regarded as propermathematics, in which theorems are proven rigorously.

The editors of the Bulletin of the American Mathematics Society (1994)invited a number of prominent mathematicians to respond to Jaffe andQuinn’s paper, receiving a long contribution from Thurston (1994) andfifteen shorter ones (Atiyah et al., 1994). Most of the respondents rejec-ted the idea of recognizing a separate, speculative branch of mathematicalactivity. Glimm expressed the general feeling best, perhaps, when he wrotethat if mathematics is to cope with the “serious expansion in the amountof speculation” it will have to adhere to the “absolute standard of logicallycorrect reasoning [which] was developed and tested in the crucible of his-tory. This standard is a unique contribution of mathematics to the cultureof science. We should be careful to preserve it, even (or especially) whileexpanding our horizons” (p. 184).

There was agreement that intuition, speculation and heuristics are veryuseful in the preliminary stages of obtaining mathematical results. Indeed,Schwartz (Atiyah et al., 1994) proposed that mathematicians abandon theold bias that dictates that only rigorous results are accepted for publication,preferring to see scholars “sometimes acting as rigorous mathematicians(if possible), sometimes writing heuristic papers (if rigorous methods donot work)” (p. 199). But all agreed that it is imperative to make a cleardistinction between a correct proof and a heuristic argument, and that thevalidity of mathematical results ultimately rests on proof.

EXPLORATION VS. PROOF

The availability in the classroom of software with dynamic graphing cap-abilities has given a new impetus to mathematical exploration, and in par-ticular has brought a welcome new interest in the teaching of geometry.Geometer’s Sketchpad (Jackiw, 1991) and Cabri Geometry (1996), for ex-ample, help students understand propositions by allowing them to performgeometric constructions with a high degree of accuracy. This makes iteasier for them to see the significance of propositions. Students can alsoeasily test conjectures by exploring given properties of the constructionsthey have produced, or even ‘discover’ new properties. The Sketchpadworkbook discusses exploration under seven headings, most of them notpart of the traditional geometry curriculum: Investigation, Exploration,Demonstration, Construction, Problem, Art and Puzzle.


Figure 1.

Dynamic software has the potential to encourage both exploration andproof, because it makes it so easy to pose and test conjectures. But unfor-tunately the successful use of this software in exploration has lent supportto a view among educators that deductive proof in geometry should bedownplayed or abandoned in favour of an entirely experimental approachto mathematical justification. Mason (1991), for example, maintains thatwith dynamic software one can check a large number of cases or even, “byappeal to continuity, an infinite number of cases”, and concludes that “truthwill be ascribed to observations made in a huge range of cases exploredrapidly on a computer” (p. 87).

An example will show how the capabilities of dynamic software couldmove some to question the need for analytical proof. Suppose a studentwants to ‘prove’ the theorem that in any triangle the perpendicular edgebisectors intersect at a single point. The student could, on paper, constructa triangle and its three perpendicular bisectors and show that this is true.But carrying out this construction with Cabri Geometry or Geometer’sSketchpad has an important advantage. It allows the student to grab apoint on the triangle and pull the triangle over the screen in such a man-ner that it changes its shape. As this is done, the perpendicular bisectorsare continuously redrawn correctly. This shows the student that the threeperpendicular bisectors still intersect at a single point, called the circum-centre of triangle, no matter what the shape of the triangle (Figure 1). Theprocedure is at least equivalent to drawing a large number of triangles onpaper, or imagining that one had drawn them.

Such a powerful feature provides the student with strong evidence thatthe theorem is true (and reinforces the value of exploration in general ingiving students confidence in a theorem). As Mason (1991) put it, it helpsthe student form a mental image. It would only be natural if the studentwere to jump to the conclusion that this exploration is entirely sufficientto establish that the perpendicular bisectors always intersect in a single

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point. Perhaps less understandable, however, is that some educators haveleapt to the same conclusion, misinterpreting the power of computers indemonstration as an indication that proof is no longer a central aspect ofmathematical theory and practice.

This is not a new issue. Exploration was an important facet of mathem-atical practice long before computers were invented, and was not seen asinconsistent with the view of mathematics as an analytic science or withthe central role of proof. What we really need to do, of course, is not toreplace proof by exploration, but to make use of both.

This is clear, first of all, when one considers that mathematical ex-ploration itself, with or without the aid of a computer, makes much useof deductive reasoning, the very foundation of proof. Polya (1957) hasdiscussed in some detail the role of deductive reasoning in exploration andproblem solving. He points out that solving a problem amounts to findingthe connection between the data and the unknown, and for this one mustuse what he calls a ‘heuristic syllogism’, a kind of reasoning that usesdeduction, in addition to circumstantial, inductive and statistical evidence.

In the second place, it is a simple fact that, while exploring and provingare separate activities, they are complementary and reinforce each other.Not only are they both part of problem solving in general, they are bothneeded for success in mathematics in particular. Exploration leads to dis-covery, while proof is confirmation. Exploration of a problem can leadone to grasp its structure and its ramifications, but cannot yield an expli-cit understanding of every link. Thus exploration can lead to conclusionswhich, though precisely formulated, must remain tentative. Though thetruth of a proposition may seem apparent from exploration, it still needs,as Giaquinto (1994) points out, ‘demonstrable justification’. Only a proof,by providing a derivation from accepted premises, can provide this.

The teacher’s classroom challenge is to exploit the excitement and en-joyment of exploration to motivate students to supply a proof, or at least tomake an effort to follow a proof supplied. One reason to go this extra step isthat exploration does not reflect the totality of mathematics itself, becausemathematicians aspire to a degree of certainty that can only be achieved byproof. A second reason is that students should come to understand the firstreason: As most mathematics educators would still agree, students need tobe taught that exploration, useful as it may be in formulating and testingconjectures, does not constitute proof.


VISUALIZATION AND VISUAL PROOFS

A number of mathematicians and logicians are now investigating the useof visual representations, and in particular their potential contribution tomathematical proofs. In the last decade or so such investigations havegained in scope and status, in part because computers have increased thepossibilities of visualization so greatly. Such studies are being pursued atmany places, such as the Visual Inference Laboratory at Indiana Universityand the Centre for Experimental and Constructive Mathematics (CECM)at Simon Fraser University in British Columbia. At most of these insti-tutions, the departments of philosophy, mathematics, computer scienceand cognitive science cooperate in research projects devoted to developingcomputational and visual tools to facilitate reasoning.

Researchers who recommend the use of visual representations in math-ematics and mathematics teaching realize, of course, that misleading dia-grams abound. Brown (1999) has presented some of the well-known ex-amples of diagrams that might lead to error. This fact alone, however,does not give reason to believe that visualization does not have promisefor investigation and teaching.

A key question raised by the intensified study of visualization is whether,or to what extent, visual representations can be used, not only as evidencefor a mathematical statement, but also in its justification. Diagrams andother visual aids have long been used to facilitate understanding, of course.They have been welcomed as heuristic accompaniments to proof, wherethey can inspire both the theorem to be proved and approaches to theproof itself. In this sense it is well accepted that a diagram is a legitim-ate component of a mathematical argument. Every mathematics educatorknows that diagrams and other visual representations are also an essen-tial component of the mathematics curriculum, where they can conveyinsight as well as knowledge. They have not been considered substitutesfor traditional proof, however, at least until recently. Today there is muchcontroversy on this topic, and the question is now being explored by severalresearchers.

According to Francis (1996), for example, the fact that more and moremathematicians turn to computer graphics in mathematical research doesnot obviate the need for rigour in verifying the knowledge acquired throughvisualization. He does recognize that “the computer-dominated informa-tion revolution will ultimately move mathematics away from the sterileformalism characteristic of the Bourbaki decades, and which still dom-inates academic mathematics”. But he adds that it would be absurd toexpect computer experimentation to “replace the rigour that mathematics

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has achieved for its methodology over the past two centuries”. For Francis,visual reasoning is clearly not on a par with sentential reasoning.

Other researchers have come to similar conclusions. Palais (1999), forexample, is a mathematician at Brandeis University who has been work-ing on a mathematical visualization program called 3D-Filmstrip for morethan five years. He reports on his use of computers to model mathematicalobjects and processes, where he defines a process as “an animation thatshows a related family of mathematical objects or else an object that arisesby some procedure naturally associated to another object” (p. 650). Heobserves that visualization through computer graphics makes it possiblenot only to transform data, alter images and manipulate objects, but alsoto examine features of objects that were inaccessible without computers.Palais concludes that visualization can directly show the way to rigorousproofs, but he stops short of saying that visual representations could beaccepted as legitimate proofs in themselves.

Borwein and Jörgenson of CECM, as well, have examined the role ofvisualization in reasoning in general and in mathematics in particular. Thetwo questions they posed to themselves were: “Can it contribute directly tothe body of mathematical knowledge?” and “Can an image act as a form of‘visual proof’?” They answer both these questions in the positive, thoughthey would insist that a visual representation meet certain qualifications ifit is to be accepted as a proof.

Borwein and Jörgenson cite the many differences between the visualand the logical modes of presentation. Whereas a mathematical proof, asa sequence of valid inferences, has traditionally been presented in senten-tial mode, a visual representation purporting to constitute a ‘visual proof’would be presented as a static picture. They point out that such a picturemay well contain the same information as the traditional sentential present-ation, but would not display an explicit path through that information andthus, in their opinion, would leave “the viewer to establish what is im-portant (and what is not) and in what order the dependencies should beassessed”.

For this reason these researchers believe that successful visual proofsare few and far between, and tend to be limited in their scope and gener-alizability. They nevertheless concede that a number of compelling visualproofs do exist, such as those published in the book Proofs without words(Nelsen, 1993). As one example, they present the following heuristic dia-gram, which proves that the sum of the infinite series 1/4 + 1/16 + 1/64 +. . . = 1/3 (See Figure 2).

Borwein and Jörgenson suggest three necessary (but not sufficient) con-ditions for an acceptable visual proof:


Figure 2.

• Reliability: That the underlying means of arriving at the proof arereliable and that the result is unvarying with each inspection

• Consistency: That the means and end of the proof are consistent withother known facts, beliefs and proofs

• Repeatability: That the proof may be confirmed by or demonstratedto others

One might wonder whether these criteria would not apply to proofs in gen-eral, not only to visual ones. One might also object that the first criterionin particular, lacking as it does a definition of ‘reliable means’, does notprovide sufficient specific guidance in separating acceptable from unac-ceptable visual proofs. Indeed, Borwein and Jörgenson make no claim tohave answered this question definitively. Nevertheless, they would assignto visual reasoning a greater role in general in mathematics, and believethat some visual representations can constitute proofs.

For other researchers, too, the idea that visual representations are nomore than heuristic tools is a dogma that needs to be challenged. Barwiseand Etchemendy (1991, 1996) sought ways to formalize diagrammaticreasoning and make it no less precise than traditional deductive reason-ing. They acknowledge that the age-old notion of proof as a derivation,consisting of a sequence of steps leading from premises to conclusion byway of valid reasoning, and in particular the elaboration of this notionin modern mathematical logic, have contributed enormously to progressin mathematics. They claim, however, that the focus on logical structuresand sentential reasoning has led to the neglect of many other forms ofmathematical thinking, such as diagrams, charts, nets, maps, and pictures,that do not fit the traditional inferential model. They also argue that it is

18 GILA HANNA

possible to build logically sound and even rigorous arguments upon suchvisual representations.

These two researchers proceeded from what they call an informationalperspective, building upon the insight that inference is “the task of ex-tracting information implicit in some explicitly presented information”(Barwise and Etchemendy, 1996, p. 180). This view leads to a criterion forthe validity of a proof in the most general sense: “As long as the purportedproof really does clearly demonstrate that the information represented bythe conclusion is implicit in the information represented by the premises,the purported proof is valid” (p. 180). The authors go on to point outthat whenever “there is structure, there is information”, and that a visualrepresentation, which may be highly structured, can carry a wealth of in-formation very efficiently. Because information may be presented in bothlinguistic and non-linguistic ways, they conclude that strict adherence toinference through sentential logic is too restrictive, inasmuch as sententiallogic applies solely to linguistic representations.

The question is how to extract the information implicit in a visual rep-resentation in such a manner as to yield a valid proof. Barwise and Etche-mendy show examples of informal derivations, such as the use of Venndiagrams, and suggest that perfectly valid visual proofs can be built ina similar fashion upon the direct manipulation of visual objects. Unfor-tunately, as they point out, the focus on sentential derivation in modernmathematics has meant that little work has been done on the develop-ment of protocols for derivation using visual objects, so that there is muchcatching up to do if visual proof is to realize its considerable potential.

Though the view of these researchers is that proof does not depend onsentential representation alone, they do not believe that visual and senten-tial reasoning are mutually exclusive. On the contrary, much of their workhas been aimed at elaborating the concept of ‘heterogeneous proof’. Build-ing upon this position, Barwise and Etchemendy (1991) have developedHyperproof, an interactive program which facilitates reasoning with visualobjects. It is designed to direct the attention of students to the content ofa proof, rather than to the syntactic structure of sentences, and teacheslogical reasoning and proof construction by manipulating both visual andsentential information in an integrated manner. With this program, proofgoes well beyond simple inspection of a diagram. A proof proceeds on thebasis of explicit rules of derivation that, taken as a whole, apply to bothsentential and visual information.


EPISTEMOLOGICAL PITFALLS

As the four research papers in this issue clearly show, dynamic geometrysoftware can be used to enhance the role of heuristics, exploration andvisualization in the classroom. The use of such approaches has alwaysposed two questions that are really a single issue: How can these ap-proaches best be integrated into the mathematics curriculum as a whole,and how can they best be used to promote understanding? The advent ofthe attractive and engaging techniques offered by computer software onlyraises the profile of this issue and the urgency with which it needs to beaddressed.

In attempting to think about the issue, it is revealing to examine theepistemology of mathematics implied by much of present classroom prac-tice and to compare it with the epistemology, as unformed and uncertainas it may be, that students bring to the classroom. As Hanna and Jahnke(1999) have discussed, the discrepancies can have significant educationalimplications.

Students are often taught, for example, that the angle-sum theorem fortriangles is true in general only because it has been proven mathematically.There is no reference to the measurement of real triangles. This practiceimplies a very specific and limited view of the nature of mathematicsand in particular of its relationship to the outside world. Students do notshare this view, however; they typically come to class with the belief thatgeometry has something to say about the triangles they find around them.For this reason it should come as no surprise to educators when studentsmisinterpret the teacher’s assertion that mathematical proof is sufficient ingeometry to mean that empirical truth can be arrived at by pure deduction.

It has been observed, to cite another example of the implications ofepistemological confusion, that students, having been shown the proof ofa theorem, will quite often ask for empirical testing, even though they saythey understand the proof (Fischbein, 1982). From a purely mathematicalviewpoint such a request seems unreasonable, and teachers usually take itas an indication that the students do not really understand what a mathem-atical proof is. From the viewpoint of an experimental scientist, however,it seems quite natural. No physicist, for example, would accept a fact astrue on the basis of a theoretical deduction alone. Thus a consideration ofthe role of mathematical proof in the experimental sciences may well shedlight on how students view proof.

Computer-supported heuristics, exploration and visualization can bevaluable tools for fostering understanding. But with the increased use ofsuch techniques in the classroom, there is also increased potential for mis-

20 GILA HANNA

understanding, with epistemology at its root. The challenge for educat-ors is to convey very clearly to students the interplay of deduction andexperimentation and the relationship between mathematics and the realworld.

To do so, educators will need to gain more insight into the assumptionsof their students. They will need to examine, for example, how studentsapproach proof while using dynamic geometry software such as CabriGeometry or the Sketchpad, which allow explorative work similar to that ofexperimental physics. But they may need to reassess their own assumptionsas well, asking in particular whether these assumptions truly reflect theaccounts of the nature of mathematics that are implicit in the practice ofmathematics itself and espoused by mathematicians and philosophers ofmathematics. Only through such a reassessment will they be in a positionto cope with the essentially epistemological questions that are bound to becreated in students’ minds when they construct mathematical proofs usingconcepts and arguments derived from their own preliminary explorations.

The four papers that follow present the results of empirical researchwhich demonstrates that the judicious use of dynamic geometry softwarein heuristics, exploration and visualization can foster an understanding ofproof. Mariotti looks at how the students’ view of geometry moves from an‘intuitive’ one, in which it is seen as a collection of evident properties, toa ‘theoretical’ one, in which it is seen as a system of related statementsthat are validated by proof. According to her, this transition is greatlyfacilitated by the use of dynamic software that affords visualization (a‘by eye’ strategy, as she calls it), exploration and in particular the use ofheuristics. The latter starts with revisiting and manipulating drawn objectsand leads to conjectures, discussion and finally to a mathematical proof.In Mariotti’s view, the dynamic software contributes to the understandingof ‘theoretical’ geometry by providing a ‘semiotic mediation’ that helpsstudents make sense of the process of exploring, conjecturing, and arguingas a way of arriving at a valid proof.

Jones’ paper focuses on the evolution of students’ ability to make use ofprecise language and to classify correctly a family of quadrilaterals. Start-ing with a visual exploration of the similarities and differences betweenquadrilaterals, the classroom experiment aims to provide a chance for thestudents to engage in deductive reasoning to arrive at an understandingof the relationships between the various properties of quadrilaterals. Thestudents then display their new understanding by creating a hierarchicalclassification for quadrilaterals. Jones observes that the dynamic softwareclearly helped students to formulate reasonably precise statements about


properties and relationships and to carry out correct deductions – bothimportant steps in constructing proofs.

Marrades and Gutiérrez examine ways in which dynamic geometrysoftware can be used to improve students’ understanding of the nature ofmathematical proof and to improve proof skills. They report on the pro-gress made by two pairs of students in justifying mathematical statements,moving from informal reasons to proper mathematical proofs by adoptingincreasingly sophisticated modes of reasoning. The authors also show howdynamic geometry software, in affording students greater access to explor-ation, heuristics and visualization, actually increased their understandingof the limitations of such informal approaches and thus of the need fordeductive proof.

Hadas, Hershkowitz and Schwartz investigate the ability of students toreflect and offer valid arguments when confronted with surprise and uncer-tainty. The students taking part in the study found a contradiction betweentheir own conjectures on the sum of the exterior angles of a polygon andthe results they obtained through measurement. It was observed that thesestudents were able to arrive at a resolution of the contradiction by usingdynamic geometry software to help them work through heuristic strategies,explorations and visual arguments.

All four papers describe the successful use of dynamic software. Itshould be noted that in all four cases this use was accompanied by care-fully designed tasks, by professional teacher input, and by opportunitiesfor students to notice details, to conjecture, to make mistakes, to reflect,to interpret relationships among objects, and to offer tentative mathemat-ical explanations. It seems reasonable to assume that the use of dynamicsoftware in the classroom might not be as effective in the absence of thesesupporting factors.

ACKNOWLEDGEMENTS

Preparation of this paper was supported in part by NATO under a Collabor-ative Research Grant, and by the Social Sciences and Humanities ResearchCouncil of Canada.

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22 GILA HANNA

Balacheff, N.: 1997, Preuve: International Newsletter on the Teaching and Learning ofMathematical Proof, in http://www-didactique.imag.fr/preuve/

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Barwise, J. and Etchemendy, J.: 1996, ‘Heterogeneous logic’, in G. Allwain and J. Barwise(eds.), Logical Reasoning with Diagrams, Oxford University Press, New York, pp. 179–201.

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American Mathematical Society 43(12), 1531–1537.

Ontario Institute for Studies in Education,University of Toronto

MARIA ALESSANDRA MARIOTTI

INTRODUCTION TO PROOF: THE MEDIATION OF A DYNAMICSOFTWARE ENVIRONMENT

ABSTRACT. This paper, which reports on a long-term teaching experiment carried out inthe 9th and 10th grades of a scientific high school, is part of a larger co-ordinated researchproject. The work constitutes a ‘research for innovation’, in which action in the classroomis both a means and result of a study aimed at introducing pupils to theoretical thinking andat studying the ways in which this process is realised. The purpose of the study is to clarifythe role of a particular software, Cabri-géomètre, in the teaching/learning process. Assum-ing a Vygotskian perspective, attention is focussed on the social construction of knowledgeand on the semiotic mediation accomplished through cultural artefacts; the functioning ofspecific elements of the software will be described and analysed as instruments of semioticmediation used by the teacher in classroom activities.

1. INTRODUCTION

This paper analyses a long-term teaching experiment carried out in the9th and 10th grades (15–16 years) of a scientific high school and aimed atintroducing pupils to theoretical thinking. The experiment is part of a co-ordinated research work concerning the introduction to theoretical thinkingat different age levels (Mariotti et al., 1997).

The research project I am presenting is rooted in the theoretical frame ofa Vygotskian perspective, with particular regard to the social constructionof knowledge and semiotic mediation accomplished through cultural arte-facts. As explained in the following sections, in this experiment, besidesthe spontaneous forms of social interaction, there are specific forms ofcontrolled and planned classroom verbal interaction realised by means of‘Mathematical Discussion’ (Bartolini Bussi, 1999, 1996).

In fact, the analysis of the teaching experiment is aimed at discussingthe specific role played by a dynamic geometry software, with particularregard to the characteristics which make it possible to introduce pupils totheoretical thinking. In particular, the paper will be devoted to discussingon the process of semiotic mediation related to the emergence of the mean-ing of proof, strictly related to the meaning of Theory. (i.e what we havecalled Mathematical Theorem (Mariotti et al., 1997)).


26 MARIA ALESSANDRA MARIOTTI

We shall limit ourselves to the description of some elements of the soft-ware, in order to present an analysis of the process of semiotic mediation(Vygotskij, 1978) that can be realised in classroom activities.

The following sections (§2-§3) are devoted to clarifying the theoreticalframe; in particular, the notion of ‘Field of Experience’, and ‘MathematicalDiscussion’ will be discussed in relation to their utilisation both in thedesign and analysis of the experimental project.

The results will be presented in terms of educational goals and analysisof the teaching-learning process, since the research project is to be con-sidered a ‘research for innovation’, in which action in the classroom is botha means and a result of the evolution of research analysis (Bartolini Bussi,1994, p. 1) On the one hand pupils achieved a theoretical perspective in thesolution of construction problems; the theoretical meaning of geometricalconstruction provided the key to accessing the general meaning of The-ory. On the other hand, the study made it possible to clarify the role of aparticular software in the teaching/learning process; the functioning of thespecific elements of the software will be described and discussed accordingto the theoretical reference frame. The software utilised is Cabri-géomètre(Baulac et al., 1988).

2. THE THEORETICAL FRAMEWORK: AN OVERVIEW

Geometrical Constructions constitute the field of experience in which class-room activities are organised. According to Boero et al. (1995, p. 153), theterm ‘field of experience’ is used to intend

the system of three evolutive components (external context; student internal con-text; teacher internal context), referred to a sector of human culture which theteacher and students can recognise and consider as unitary and homogeneous.

The development of the field of experience is realised through the socialactivities of the class; in particular, verbal interaction is realised in collect-ive activities aimed at a social construction of knowledge: i.e. ‘Mathemat-ical Discussions’, that is

polyphony of articulated voices on a mathematical object, that is one of the objects– motives of the teaching – learning activity (Bartolini Bussi, 1996, p. 16).

Polyphony occurs between the voice of practice and the voice of theory.The practice of the pupils consists in the experience of drawing, evokedby:

• concrete objects, such as drawings, realised by paper and pencil, rulerand compass.

• computational objects such as Cabri figures or Cabri commands.

THE MEDIATION OF A DYNAMIC SOFTWARE ENVIRONMENT 27

Geometry theory, imbedded in the Cabri microworld, is evoked by theobservable phenomena and the commands available in the Cabri menu.

Figures and commands may be considered external signs of the Geo-metric theory, and as such they may become instruments of semiotic medi-ation (Vygotskij, 1978), as long they are used by the teacher in the concreterealisation of classroom activity to introduce pupils to theoretical thinking.

3. THE FIELD OF EXPERIENCE OF GEOMETRICAL CONSTRUCTIONS IN

THE CABRI ENVIRONMENT

3.1. Reference culture

The reference culture is that of classic Euclidean Geometry. EuclideanGeometry is often referred to as ‘ruler and compass geometry’, becauseof the centrality of construction problems in Euclid’s work. The funda-mental theoretical importance of the notion of construction (Heath, 1956,p. 124 segg.) is clearly illustrated by the history of the classic impossibleproblems, which so much puzzled the Greek geometers (Henry, 1994).

Despite the apparent practical objective, i.e. the drawing which canbe realised on a sheet of paper, geometrical constructions have a theor-etical meaning. The tools and rules of their use have a counterpart in theaxioms and theorems of a theoretical system, so that any construction cor-responds to a specific theorem. Within a system of this type, the theoremvalidates the correctness of the construction: the relationship between theelements of the drawing produced by the construction are stated by atheorem regarding the geometrical figure represented by the drawing.

The world of Geometrical Construction has a new revival in the dy-namic software Cabri-géomètre. As a microworld (Hoyles, 1993), it em-bodies Euclidean Geometry; in particular, as it is based on the intersectionsbetween straight lines and circles, Cabri refers to the classic world of ‘rulerand compass’ constructions.

However, compared to the classic world of paper and pencil figures,the novelty of a dynamic environment consists in the possibility of directmanipulation of its figures and, in the case of Cabri, such manipulationis conceived in terms of the logic system of Euclidean Geometry. Thedynamics of the Cabri-figures, realized by the dragging function, preservesits intrinsic logic, i.e. the logic of its construction; the elements of a figureare related in a hierarchy of properties, and this hierarchy corresponds to arelationship of logic conditionality.

Because of the intrinsic relation to Euclidean geometry, it is possibile tointerpret the control ‘by dragging’ as corresponding to theoretical control


– ‘by proof and definition’ – within the system of Euclidean Geometry. Inother words, it is possible to state a correspondence between the world ofCabri constructions and the theoretical world of Euclidean Geometry.

Advantages and pitfalls of using Dynamic Geometry Environments1

have been largely discussed in the literature. On the one hand, the pos-sibility of making an explicit distinction between ‘drawing’ and ‘figure’has been pointed out (Laborde, 1992); on the other hand, many studieshave shown that students’ interpretation of the drag mode is not obviousand cannot be taken for granted (Straesser, 1991; Noss et al., 1993; Hoelz,1996). In the case of the Geometer’s sketchpad, Goldenberg and Cuoco(1998) present and discuss alternative interpretations of the observablephenomena, generated by dragging on the screen.

As explained in the following, in our project, the development of theField of Experience is based on the potential correspondence betweenCabri construction and Geometric theorems. Once a construction problemis solved, i.e. if the Cabri-figure passes the dragging test, a theorem canbe proved with a geometric proof. Thus, solving construction problems inthe Cabri environment means accepting not only all the graphic facilitiesof the software, but also accepting a logic system in which its observablephenomena will make sense. The explicit introduction of this interpretationand the continuous reference to the parallel between Cabri enviroment andGeometry theory constitutes the basis of our teaching project.

3.1.1. The theoretical dimension of mathematical knowledgeAlthough it is important to distinguish between the intuitive constructionof knowledge and its formal systematisation, one must recognise that thedeductive approach, primarily set up in the Euclid Elements, has becomeinherent in mathematical knowledge. Even when stress is placed on theheuristic processes, it is impossible to neglect the theoretical nature ofmathematics, as the following passage, quoted by Schoenfeld (1994), cla-rifies.

What does mathematics really consist of? Axioms (such as the parallel postu-late)? Theorems (such as the fundamental theorem of algebra)? Proofs (such asGödel’s proof of undecidability)? Definitions (such as the Menger definition ofdimension)? [. . .] Mathematics could surely not exist without these ingredients;they are essential. It is nevertheless a tenable point of view that none of them isthe heart of the subject, that mathematician’s main reason for existence is to solveproblems and that, therefore, what mathematics really consists of is problems andsolutions. (Halmos, 1980, p. 519)

As often pointed out, the distance between the theoretical and the intuitivelevel raises great difficulties (Fischbein, 1987), but theoretical organisationaccording to axioms, definitions and theorems, represents one of the basic


elements characterising mathematical knowledge. In particular, as far astheorems are concerned, it is worth reminding (Mariotti et al., 1997) thatany mathematical theorem is characterised by a statement and a proof andthat the relationship between statement and proof makes sense within aparticular theoretical context, i.e. a system of shared principles and in-ference rules. Historic – epistemological analysis highlights important as-pects of this complex link and shows how it has evolved over the centuries.The fact that the reference theory often remains implicit leads one to forgetor at least to underevaluate its role in the construction of the meaning ofproof. For this reason is seems useful to refer to a ‘mathematical theorem’as a system consisting of a statement, a proof and a reference theory.

3.2. The external context

The external context is determined by the ‘concrete objects’ of the activ-ity (paper and pencil; the computer with the Cabri software; signs – e.g.gestures, figures, texts, dialogues).

In the following sections of this paper, we shall focus on the partic-ular ‘objects’ offered by the Cabri environment, which however must beconsidered in a dialectic relationship with all the other objects available.

The activity starts by revisiting drawings and artefacts which belong tothe pupils’ experience. Such objects are part of physical experience, thecompass being a concrete object the use of which is more or less familiarto the pupils. In any case, the students are familiar with the constraints andrelationships which determine possible actions and expected results; forinstance, the intrinsic properties of a compass directly affect the propertiesof the graphic trace produced.

Revisitation is accomplished by transferring the drawing activity intothe Cabri environment, so that the external context is moved from the worldof ruler and compass drawings to the virtual world of Cabri figures andcommands.

In a software environment the new ‘objects’ available are EvocativeComputational Objects (Hoyles, 1993; Hoyles and Noss, 1996, p. 68),characterised by both their own computational nature and the evocativepower caused by their relationship with geometrical knowledge:

• the Cabri-figures realising geometrical figures;• the Cabri-commands (primitives and macros), realising the geomet-

rical relationships which characterise geometrical figures;• the dragging function which provides a perceptual control of the cor-

rectness of the construction, corresponding to the theoretical controlconsistent with geometry theory.


The development of the field of experience is based on the practice carriedout through activities in the context of Cabri: construction tasks, interpreta-tion and prediction tasks and mathematical discussions. Such developmenthowever, also concerns the practice of ruler and compass constructions,which become both concrete referents and signs of the Cabri figures. Therelationship between drawings (accomplished on paper, using ruler andcompass) and Cabri figures constitutes a peculiar aspect of the externalcontext, which presents a double face, one physical and the other virtual.

3.3. The internal context of the pupil

3.3.1. Intuitive and deductive geometryTraditionally in the Italian school, deductive geometry is introduced atthe beginning of the 9th grade (entrance to high school). In the previousgrades geometry is studied, but usually at an ‘intuitive’ level: this meansthat Geometry is presented to pupils as a collection of ‘definitions’, namingand describing geometrical figures, and ‘facts’ stating particular properties.Most of these facts have a high degree of evidence, and in any case thearguments eventually provided by the teacher have the specific aim of con-structing such evidence. Moreover, pupils are never asked to justify theirknowledge, the truth of which is considered immediate and self-evident,i.e. intuitive (Fischbein, 1987). As a consequence, at the beginning of highschool, pupils generally have an intuitive geometrical background whichmust be reorganised according to a deductive approach.

The delicate relationship between intuitive knowledge and its theoret-ical systematisation is usually very difficult to manage: pupils fail to graspthe new with respect to the old. Actually, it is very difficult to understandwhy well known properties should be put into question and long argumentsused to support their truth, which is so evident.

In the pupils’ experience, justifying is the prerogative of the teacherand, rather than providing pupils with a basis for a deductive method, itaims at convincing one of the ‘evidence’ of a certain fact. However, whena statement reaches the status of evidence, any argument becomes uselessand ready to be forgotten. According to its nature, intuition (Fischbein,1987) contrasts the very idea of justification, and in this respect, intuitivegeometrical knowledge may constitute an obstacle to the development ofa theoretical perspective.

In summary, when the deductive approach is introduced, a crucial pointis that of changing the status of justification.

A deductive approach is deeply rooted in the practice of justification (deVilliers, 1990; Hanna, 1990). Proving consists in providing both logicallyenchained arguments which are referred to a particular theory, and an ar-


gumentation which can remove doubts about the truth of a statement. Thistwofold meaning of proof is unavoidable and pedagogically consistent(Hanna, 1990). It is often pointed out and commonly accepted, however,that arguing and proving do not have the same nature; arguing has the aimof convincing, but the necessity of convincing somebody does not alwayscoincide with the need to state the theoretical truth of a sentence.

. . .il y a une très grande distance cognitive entre le fonctionnement d’un raison-nement qui est centré sur les valeurs épistémique liées à la compréhension ducontenu des propositions et le fonctionnement d’un raisonnement centré sur lesvaleurs épistémique lieées au statut théorique des propositions.

(Duval, 1992–93, p. 60)

The distance between these two modalities explains the reason why ar-guing can often become an obstacle to the correct evolution of the veryidea of proof (Balacheff, 1987; Duval, 1992–93).

To sum up, in the case of a deductive approach, two interwoven aspectsare involved: on the one hand, the need for justification and on the otherhand the idea of a theoretical system within which that justification maybecome a proof.

3.4. The internal context of the teacher

The presence of the computer and of particular software certainly rep-resents a perturbation element in the internal context of the teacher. Theteacher has to elaborate a new relationship to mathematical knowledge,augmented by the whole set of relations which link it to the computer ingeneral and cabri in particular. At the same time, the teacher has to adapthis/her role of mediator taking into account the new elements offered bythe software.

Specific issues arise that are related to the nature of the mediationinstruments, with particular regard to problems referring to the ‘compu-tational transposition of knowledge’ (Balcheff and Sutherland, 1994; Bal-cheff, 1998), i.e. resulting from implementation in a computer.

Up until now, no specific analysis of these aspects has been carriedout, although there is some evidence of an interesting development in theteachers’ conceptions, which will require further investigation.


4. THE ACTIVITIES

4.1. Construction activity

According to the general hypothesis of our project, construction problemsconstitute the core of the activities proposed to the pupils. The main object-ive was the development of the meaning of ‘construction’ as a theoreticalprocedure which can be validated by a theorem.

The construction task was articulated as follows:

a) providing the description of the procedure used to obtain the reques-ted figure;

b) providing a justification of the correctness of that procedure.

Pupils worked in pairs, and each pair had to provide a written text drawnup in common. These written reports constituted the basis on which thecollective discussions were organised.

Within the Cabri environment, as soon as the dragging test is accepted,there is a need to justify one’s own solution. The necessity of a justificationfor the solution comes from the need to explain why a certain constructionworks (that is, passes the dragging test). Such a need is reinforced duringthe collective discussion, when different solutions are compared, by val-idating one’s own construction in order to explain why it works and/or toforesee whether or not it will function.

Experimental evidence (Mariotti, 1996) shows that, as far as the paperand pencil environment is concerned, the theoretical perspective is veryhard to grasp. When a drawing is produced on a sheet of paper, it is quitenatural for the validation of the construction to be focussed on the drawingitself and on a direct perception. Ambiguity between drawings and figuresmay represent an obstacle: although the question is about the drawing, itssense concerns the geometrical figure it represents.

A theoretical validation concerns the construction procedure rather thanthe drawing produced: each step of the procedure corresponds to a geo-metrical property, and the entire set of properties given by the procedureconstitutes the hypothesis of a theorem proving the correctness of theconstruction.

When a construction problem is presented in the Cabri environment thejustification of the correctness of a solution figure requires an explanationof why some constructions function and others do not. This implies shift-ing of the focus from the drawing obtained to the procedure that producedit. The intrinsic logic of a Cabri-figure, expressed by its reaction to thedragging test, induces pupils to shift the focus onto the procedure, and indoing so it opens up to a theoretical perspective.


4.2. ‘Mathematical Discussion’ activities

Among the classroom activities, ‘Mathematical discussions’ (BartoliniBussi, 1991, 1999) take up an essential part in the educational process, withspecific aims, which are both cognitive (construction of knowledge) andmetacognitive (construction of attitudes towards learning mathematics).

In the literature, the role of discussion in the introduction to the idea ofproof has been analysed. Both aspects of continuity and rupture have beenclearly described; the most radical opinion is held by Duval, who claimsthat

Passer de l’argumentation à un raisonnement valide implique une décentrationspécifique qui n’est pas favorisée par la discussion ou par l’interiorisation d’unediscussion. (Duval, 1992–93, p. 62)

Our proposal refers to a specific type of discussion, mathematical discus-sion, which is not a simple comparison of different points of view, nota simple contrast between arguments. The main characteristic (BartoliniBussi, 1999) of this kind of discussion is the cognitive dialectics betweenpersonal senses (Leont’ev, 1976/1959, p. 244) and general meaning, whichis constructed and promoted by the teacher. In this case, the cognitive dia-lectics takes place between the sense of justification and general meaningof mathematical proof.

Different senses of justification correspond to possible different goalsof the discussion, whereas moving from one goal to another correspondsto the evolution of the sense of justification, which is the main motive ofdiscussion.

The role of the teacher is fundamental, in order to direct the goal ofthe discussion and to guide the evolution of personal senses towards thegeometrical meaning of a construction problem, and more generally to thetheoretical perspective.

This corresponds to two different types of motivation which determinethe discussion activity and interrelate and support each other.

On the one hand, justifications must be provided within a system ofshared principles.

On the other hand, the methods of validation must be shared and therules of mathematical argumentation made explicit.

In the following two sections, after a brief general summary of theresults coming from the experimental data, we shall discuss the processof semiotic mediation with respect to the evolution of the pupils’ internalcontext. Some examples will be analysed, drawn from the transcripts ofcollective discussions and from the written reports provided by the pupils.


5. EVOLUTION OF THE INTERNAL CONTEXT

The analysis of the experimental data highlights the development of themeaning of justification emerging for both the construction tasks and theMathematical Discussions; such evolution may be broken up into a se-quence of steps.

a) Description of the solutionFrom the very beginning the pupils are asked to provide a writtenreport of the solution process. The main objective consists in commu-nicating to the teacher and the classmates one’s own reasoning; as aconsequence, the main objective in writing the report is that of beingunderstood.

b) Justifying the solutionThe solutions described in the reports are proposed by the teacher andform the basis of a collective discussion. Once the pupils have accep-ted the criterion of validation by dragging, attention is shifted fromthe product (the drawing) to the procedure followed to obtain it; thusthe main intention in writing the reports becomes that of providinga justification for that procedure. After the first discussions, a newobjective emerges: the reports must provide a clear description of theconstruction, but also a justification so that it can pass the collectivetrial, and this means providing good arguments which can make thesolution acceptable.

c) Justifying according to shared and stated rulesIn the collective discussions the main purpose is the defence of one’sown construction, and for this reason it becomes necessary to statea number of rules to be respected; in other words an agreement onthe acceptable operations must be negotiated in class. This may beconsidered the origin of a theoretical perspective: looking for a justi-fication within the system of rules introduces pupils to a theoreticalstatus of justification.

The collective discussion guided by the teacher according to a specificobjective determines the meaning of the construction problem to be de-veloped. In this process a basic role is played by Cabri, and the followingsections will be devoted to discussing the process of semiotic mediation,accomplished by the teacher through the use of specific software elements.

5.1. Semiotic mediation

Historic and epistemological analysis confirms the productive interactionbetween theory and practice in the development of mathematical know-


ledge (Bartolini and Mariotti, 1999). A crucial element of the dialecticrelationship between theory and practice is represented by technical tools.

Tools have a twofold function, the former, externally oriented, is aimedat accomplishing an action; the latter, internally oriented, is aimed at con-trolling the action. This distinction, commonly used in studies about thepractice and development of technologies (e. g. Simondon, 1968), canbe elaborated starting from the seminal work of Vygotskij (1978), whointroduces the theoretical construct of semiotic mediation. Vygotskij dis-tinguishes between the mediation function of what he calls tools2 and signs(or instruments of semiotic mediation). Both are produced and used byhuman beings and are part of the cultural heritage of mankind. Althoughassumed in the same category of mediators, ‘signs and tools’ are clearlydistinguished by Vygotskij (1978, p. 53).

The basic analogy between sign and tools rests on the mediating function thatcharacterises each of them. They may, therefore, from the psychological perspect-ive, be subsumed under the same category. . . . of indirect (mediated) activity.

(ibd. p. 54)

However, their function cannot be considered isomorphic; the differencebetween the two elements rests on “the different way that they orient thehuman behaviour”. (ibd. p. 54). The tool function, externally oriented, isto serve as conductor of the human activity aimed at mastering nature. Thesign function, internally oriented, is a means of internal activity aimed atmastering oneself.

The use of the term psychological tools, referred to signs internallyoriented, is based on the analogy between tools and signs, but also on therelationship which links specific tools and their externally oriented use totheir internal counterpart.

According to Vygotskij, the mastering of nature and the mastering ofoneself are strictly linked, “just as man’s alteration of nature alters man’sown nature” (ibd., p. 55).

The process of internalisation as described by Vygotskij may transformtools into psychological tools: when internally oriented a ‘psychologicaltool’ will shape new meanings, thus functioning as semiotic mediator.

The example of the counting system, as discussed by Vygotskij himself(ibd., p. 127), shows the transition from external to internal orientation; asystem of counting produced and employed to evaluate quantity may beinternalised and function in the solution of problems, so as to organise andcontrol mental behaviour.

An illuminating description of such a process of internalisation in thecase of the compass and circle is given by Bartolini Bussi et al. (to appear).In this case, rather than through a historic reconstruction, the process is


described as it was accomplished by the pupils of a primary school class,during a long-term teaching experiment.

The geometric compass, embodied by the metal tool stored in every school-case,is no more a material object: it has become a mental object, whose use may besubstituted or evoked by a body gesture (rotating hands and arms).

(Bartolini Bussi et al., to appear)

5.2. The process of semiotic mediation in the Cabri environment

A microworld like Cabri is a particular case of cultural artefact, character-ised by the presence of:

• an object constructed with the aim of achieving a result (drawing im-ages on the screen), and for this reason it was designed to incorporatea certain knowledge, i.e. Euclidean Geometry;

• utilisation schemes, namely the modalities of action accomplished byusing the artefact according to a specific goal (drawing a screen imagewhich will pass the dragging test).

The dragging test, externally oriented at first, is aimed at testing percep-tually the correctness of the drawing; as soon as it becomes part of in-terpersonal activities – peer interactions, dialogues with the teachers, butespecially the collective discussions – it changes its function and becomesa sign referring to a meaning, the meaning of the theoretical correctness ofthe figure.

Many examples in the literature show the potentialities of a Computatio-nal-technology artefact (computer, microworld, graphic calculator . . .) interms of meanings construction, but at the same time the instability of theprocesses of meanings construction, related to the use of an artefact (for afull discussion see Mariotti, to appear).

The framework of Vygotskian’s theory makes it possible to overcomethis ‘impasse’.

The functioning of an artefact in the development of meaning can bedescribed taking into account the process of semiotic mediation whichdevelops at different levels:

• The pupil uses the artefact, according to certain utilisation schemes,in order to accomplish the goal assigned by the task; in so doing theartefact may function as a semiotic mediator where meaning emergesfrom the subject’s involvement in the activity.

• The teacher uses the artefact according to specific utilisation schemesrelated to the educational motive. In this case, as explained in thefollowing examples, the utilisation schemes may consist in particularcommunication strategies centred on the artefact.


Thus, the artefact is exploited by a double use, with respect to which itfunctions as semiotic mediator. On the one hand, meanings emerge fromthe activity – the learner uses the artefact in actions aimed at accomplishinga certain task; on the other hand, the teacher uses the artefact to direct thedevelopment of meanings that are mathematically consistent.

The artefact incorporates a mathematical knowledge accessible to thelearner by its use, but meaning construction requires the guidance of theteacher, who organises and directs specific activities in which the develop-ment of meanings can be recognised and accepted mathematically.

Meanings are rooted in phenomenological experience (actions of theuser and feedback from the environment, of which the artefact is a com-ponent), but their development is achieved by social construction in theclassroom, under the guidance of the teacher.

The following sections will be devoted to illustrating the process ofsemiotic mediation accomplished through the action of the teacher usingthe facilities offered by the Cabri environment.

In the rich collection of data, gathered during the teaching experiment,some examples were selected. They can be considered representative ofwhat happened in the classroom and we hope they illustrate our resultsclearly. The difficulties related to the selection of examples can be easilyimagined, since the description of a long term process taking place in a‘living’ classroom is always a hard task.

5.3. Mediation of the ‘history’ command

The first example concerns a collective discussion; the analysis, carried outon the transcript of the record, shows how within the Cabri environmentthe teacher can find specific tools of semiotic mediation contributing to thedevelopment of the meaning of geometrical construction.

The episode (for more details see Mariotti and Bartolini Bussi, 1998)involved one of the experimental classes of the project (9th grade – 15 yearsold – of a scientific high school (Liceo Scientifico)); 19 out of 23 pupils inthe class participated in the activity which constituted the very beginningof the experimentation.

The first part of the activity takes place in the Computer room, wherethe pupils sit in pairs at the machine. They have general ability with thecomputer, and are allowed to explore the software freely for about half anhour, after which they are presented with the following task.

Construct a segment.

Construct a square which has the segment as one of its sides.


Figure 1. The construction does not pass the dragging test.

The pupils are asked to realise a figure on the screen and to write down adescription both of the procedure and of their reasoning. The meaning ofthe term construction is not explained, leaving the pupils free to interpretthe task. As expected, the protocols collected contain differently obtainedsolutions, some referring to geometrical properties, others to perceptualcontrol. These solutions are differently transformed by using the drag-ging function, and such differences provide the basis for the followingdiscussion.

The first solution proposed by the teacher is that given by Group1 (Gio-vanni and Fabio): four consecutive segments, perceptually arranged in asquare.

When the teacher asks the pupils to judge this solution, everybodyagrees that control must be exerted on the particular drawing; accordingto the well known definition of square, they suggest to measure the sidesand the angles. The main elements arising from the discussion are theuse of measure and the precision related to it. The pupils are stuck: allinterventions show that the shared objective concerns the control of thedrawn square.

When the teacher drags the figure (Figure 1) everybody agrees that itis no longer a square. At this point, another solution (Group 3, Dario &Mario) is proposed by the teacher.

‘T’ (= Teacher)

21 T: Well, I’d like to know your opinion about the construction made by Darioand Mario

22 Marco: They drew a circle, then two perpendicular lines. . .23 T: Do you know where they started from?24 Michele: We can use the command ‘history’.


Figure 2. Group 3 Dario & Mario: The first steps of the construction.

25 T: Let’s do that. They took a segment, then they . . .

[The step by step construction follows (Figure 2)]They drew a line perpendicular to the segment, then the circle . . . in youropinion, what is it for ? What is its use?

SILENCE

Is there a logic in doing so, or did they do it just because they felt likedrawing a perpendicular line . . . a circle . . . Alex, what do you think? . . .

26 Alex: the measure of the segment is equal to the measure reported by the circleon the perpendicular line.

27 T: You mean that the circle is used to assure two equal consecutive segments,the first one and the other one on the perpendicular line . . . and the perpen-dicular . . .

28 Chorus: it is used . . . to obtain . . . an angle of 90◦29 T: I know that the square has an angle of 90◦ and four equal sides or three

equal angles. . . then let’s see if that is true . . . let’s go on.

Intersection between line and circle. They (Dario e Mario) determined theintersection point between the line and the circle . . . why did they need thatpoint?

30 Chiara: the intersection point between the line and the segment . . .

31 T: and what should you draw from there ?32 Chiara: a segment, perpendicular to the line


33 T: what else??34 Chorus: parallel to the segment. . .35 T: let’s see what they did . . .

Let us analyse this episode.A first suggestion comes from Michele: use the History command. Theteacher catches the suggestion and activates the command. While the firststeps of the construction are repeated, the teacher describes what has beendone, until she interrupts the description and asks the pupils to detect the‘motivations’ for these actions.

This intervention (which triggers what we call interpretation game) isaimed at provoking the first shift from the procedure to a justification ofthe procedure itself.

The move from action to explaining the motivation of this action isdifficult and the pupils are pressed (25), until Alex (26) expresses the rela-tionship between two of the segments according to the series of commandspreviously executed. The teacher (27) reformulates her statement in termsof motivations: ‘You mean that the circle is used to assure two equal con-secutive segments . . .’. The Chorus appropriates the teacher’s expression‘used. . .’ and continues in terms of motivation.

The discussion goes on: a new game is activated (which we have calledthe prediction game): the pupils are now asked to foresee the next step, tomotivate and then compare it with the step recorded in the history.

The discussion goes on, different solutions are compared, and finallytogether with the acceptance of a solution in terms of the dragging test, anew relationship to drawing is achieved. It becomes possible to explain thecorrectness of a construction controlling the ‘logic’ of the procedure.

As clearly shown by the previous analysis, the presence and availabilityin the computer of a decontextualised and detemporalised copy of the con-struction procedure allows the teacher to realise specific communicationstrategies (the interpretation game and the prediction game) consistent withthe educational motive. Without explicit comments or implicit information(gestures, and so on) towards the expected answer, the pupils’ attentionmoves from the drawing produced to the construction procedure, and at thesame time the idea of a justification for the single steps of the constructionis introduced.

It is interesting to remark that the History command remains a basicelement in the process of theorems production. The sequence of construc-tion steps may represent the temporal counterpart of the logic hierarchybetween the properties of a figure. In the dynamics of the Cabri figure therelationships between the geometric properties are expressed globally, sothat some relationships of logic dependence may be hidden and missed by


the pupils. The History command reintroduces the temporal dimension andallows one to grasp the construction process in its development; in doingso, it supports the control of the logic relationships between the propertiesinvolved.

5.4. Using commands and axioms

The idea of proof involves two kinds of strictly interconnected difficulties.It is necessary to introduce on the one hand the general idea of justification,on the other the idea of a justification with regard to specific principles andaccepted rules of inference, i.e. in order to be a proof, any justificationmust be referred to a theory.

As already discussed, the relationship between Cabri constructions andgeometrical theorems is based on the link between the logic of Cabri, ex-pressed by its commands, and the Euclidean Geometry expressed by itsaxioms, theorems and definitions. When the whole Cabri menu is avail-able, the relationships between the main concepts and properties, availablethrough the geometrical primitives of Cabri, remain largely hidden. As aconsequence, the complexity of the corresponding geometric system maybe too difficult to be managed by novices. In fact, because of the richnessof the geometrical properties available, it is difficult to state what is given(axioms and ‘old’ theorems) and what must be proved (‘new’ theorems).Generally speaking, the richness of the environment might emphasise theambiguity about intuitive facts and theorems, so that it can even representan obstacle to grasping the meaning of proof.

Thus, taking advantage of the flexibility of the Cabri environment, in-stead of providing the pupils with an already-made Cabri menu, corres-ponding to an already stated axiomatization, we decided to construct ourCabri menu ‘step by step’, in parallel with the corresponding theory.

At the beginning an empty menu was presented and the choice of com-mands discussed, according to specific statements selected as axioms. Then,as new constructions were introduced and the corresponding new theoremsenlarged the theory, new commands were added to the menu. In this waythe geometry system was slowly built up with a twofold aim. The pupilsparticipated (Leont’ev, 1976/1959) in the construction of both an axiomat-ization and a corresponding menu; at the same time, the complexity ofthe theoretical system increased at a rate which the pupils were able tomanage.

It is impossible to go into the details of the protocol analysis. Nonethe-less, the following example illustrates how the use of the commands maysupport the construction of the theoretical meaning of a justification. Thetask proposed to the pupils was the following:


Figure 3. The construction of the mid-point as realised by G. & C.

Construct the mid point of a segment.

G. and C. (9th grade – 15 years old)

I create a segment through two points. I fix three other points on the screen andconstruct an angle with them. With (the command) ‘report of an angle’, I carrythis angle on the edges of the segment and create the intersection of these tworays. Using (the command) ‘circle (centre, point)’, with the centre on the edges ofthe segment and point on the intersection of the rays, I create two equal circles.Joining the two intersections I find the mid-point O.

I did that because creating the equal angles on this segment an isosceles triangleis created. Using the equal sides of this triangle as radios of two new circles, I canconstruct two equal circles on the edges of the segment.

The protocol does not contain any drawing, and the pupils probably referto the screen image. In the figure (Figure 3) the construction has beenreproduced step by step to illustrate the procedure described by the pupils.


This protocol shows a good example of a first step in the development ofthe meaning of theoretical justification. The description of the constructionand its justification are still mixed. This shows the difficulty of separatingthe operational aspect – realising the Cabri-figure – and the theoretical as-pect which consists in identifying the geometric relationships drawn fromthe figure according to its construction. At the same time, however, therelationship between the use of the command and its theoretical mean-ing appears clearly: the protocol illustrates the process of internalisationthrough which the external signs – the Cabri commands – are transformedinto internal tools related to theoretical control.3 According to a classicaxiomatization (Hilbert, 1899/1971; Heath, 1956, p. 229), the particularcommand ‘report of an angle’ corresponds to one of the axioms introducedin the theory.

Similar examples (see for instance the following protocol) can be foundand provide evidence of the fact that the problem of construction has achie-ved a theoretical meaning. Similarly, examples can be found showing thatthe Cabri commands may function as external signs of the theoretical con-trol, corresponding to using axioms, theorems and definitions. Let us nowconsider the following example:

Construct the line perpendicular to line t and passing through itsrelative point P, belonging to it.

Sa and Si (9th grade – 15 years old)

First of all we go to ‘Creation’ and we construct line t; in ‘Creation’ we also findpoint P and we constrain it on the line. In the same way we fix two other points Aand B, so that P is located between them and we constrain them on the line. Then,in ‘Construction’, we choose the label ‘angle bisector’ of angle APB, which isalso perpendicular to line t passing through P.

Figure 4. The drawings provided by Sa & Si.


Justification

According to the definition of perpendicular line which says that a line t is per-pendicular to a line s if it is the angle bisector of a straight angle which has thevertex in a point of s, I can prove that the angle bisector of the straight angle APBis perpendicular to line t.

The explicit reference to the definition4 of ‘perpendicular line’ allows thepupils to validate the correctness of the construction, based on the use ofthe command ‘angle bisector’.

The fact that the commands available in the current menu are recog-nised as theoretical properties available in the theory, makes the construc-tion procedure itself an external sign of a theorem of the theory. The com-mand, expressed by a ‘label – word’, may function as external control, andrefers to the possibility of making that theorem explicit. In the case of theexample the command ‘angle bisector’ refers to a construction, previouslyrealised, and corresponds to a theorem included in the Theory available.

According to Vygotskij, the process of internalisation of such signsdetermines the construction of the theoretical meaning of the construc-tion problems and opens up to the theoretical perspective for geometricalproblems in general.

In conclusion, according to our main hypothesis, pupils’ introductionto contruction problems within the Cabri enviroment provided a key toaccessing a theoretical perspective. The main point was the interpretationof the constraints of the environment in terms of geometrical propertiesand in terms of the mutual dependence between geometrical properties.

As the examples should have shown, that interpretation resulted from alengthy process, carefully guided by the teacher, through specific activ-ities among which collective discussions took a fundamental part. Thefollowing section is devoted to a micro analysis of the process of semioticmedation, centred on the use of the artefact Cabri.

6. INTERNALISATION OF DRAGGING AS THEORETICAL CONTROL

Interesting aspects related to the development of the meaning of mathem-atical theorem can be highlighted in the solution of open-ended problems,i.e. the particular type of task asking for a conjecture and its proof. Infact, exploration in the Cabri environment with the aim of producing aconjecture is based on the awareness of the theoretical interpretation ofa cabri-figure. The data of the problem can be realized by constructing aCabri-figure, so that the hierarchy of the construction realises the geomet-ric relationship given by hypothesis; thus the interpretation of perceptive


invariants in terms of geometric conjecture is based on the interpretationof the dragging control in terms of logic control. In other terms the use ofCabri in the generation of a conjecture is based on the internalisation of thedragging function as a logic control, which is able to transform perceptualdata into a conditional relationship between hypothesis and thesis.

. . . the changes in the solving process brought by the dynamic possibilities ofCabri come from an active and reasoning visualisation, from what we call aninteractive process between inductive and deductive reasoning.

(Laborde and Laborde, 1991, p. 185)

Consciousness of the fact that the dragging process may reveal the rela-tionship between the geometric properties embedded in the Cabri figuredirects the ways of transforming and observing the screen image.

As a consequence, evidence of the process of internalisation of thetheoretical control can be shown by the ways in which pupils constructand transform the image on the screen, when they are solving open-endedproblems:

• the figure to be explored is constructed considering that the propertiesprovided in the hypotheses are realised by the corresponding Cabricommands;

• the conjectures may emerge from exploration by dragging, but theirvalidation is sought within the Geometric theory, i.e. in principle,conjectures ask for a proof.

The following example aims at analysing this process of internalisation.

6.1. A particular case: the rectangle problem

Let us consider the following activity, proposed to a class (9th grade – 15years old), during a session in the computer lab. The pupils sit in pairs atthe computer and are asked to perform the following task:

Draw a parallelogram, make one of its angles right and write yourobservations.

Almost all the pupils correctly (although differently) construct a parallelo-gram and describe the construction. In some cases, the drawing, providedby the pupils in the report, reproduces the screen image; in many cases,in order to reproduce the complexity of the Cabri figure – an image con-trolled by the properties /commands – the drawing is completed with la-bels referring to the construction accomplished (for instance: ‘line by twopoints’, ‘parallel line’). These elements testify to the intention of realisinga Cabri figure incorporating the hypotheses related to the parallelogram(see Figure 5 below.).


Figure 5. Use of command ‘mark and angle’ – Giovanni wrote: “After that, we markedangle A. We moved the lines DB and AC until the angle A reaches 90◦”.

Following the instructions given by the task, the pupils realise the newhypothesis (‘make an angle right’) and transform the figure by dragging,making an adjustment of the perpendicularity of the sides ‘by eye’. Theconjecture comes straightforward.

Because of the immediacy of the conclusion, the main difficulty con-cerns the production of the proof rather than the formulation of the con-jecture (all conjectures were correct but there were some errors in theproofs).

The strategies followed in this part of the solution reveal the presenceof particular ‘signs’ used in constructing the proof. Such ‘signs’ can beinterpreted as means of external control (Bartolini Bussi et al., to appear)on the logic operations required to produce the proof.

The signs are generated within the software environment and derivetheir semantics from that of the software, i.e. from the system of meaningswhich have so far emerged in the practice. Let us now analyse two of theobservable signs.

6.1.1. ‘Mark an angle’ and ‘perpendicular line’Before dragging the figure, some of the pupils use the command ‘markan angle’. This command, without which it is impossible to obtain themeasure of an angle, is used as a means to control the hypothesis of ‘one(only one) right angle’, under which the exploration must be carried outand the conjecture formulated.


Figure 6. Examples of different signs supporting the identification of the hypothesis.

In those cases, the drawings in the reports reproduce a sequence ofsnapshots (photograms), showing the different phases of the solution anddifferently marked angles (Figure 5).

It is interesting to remark that the software distinguishes between thesign for ‘mark an angle’ and the sign for ‘right angle’. The same distinctionis reproduced on the drawing.

Other pupils use a different type of ‘sign’ as follows: they construct aline passing through one of the vertices and ‘perpendicular’ to one of thesides of the parallelogram. Dragging one of the free vertices, one of theoblique sides is made to coincide with ‘the perpendicular line’. In theirreports, the pupils try to express the dynamics of the Cabri figure; forinstance pupils draw arrows connecting the vertex and the moving sideto the ‘perpendicular line’ (Figure 6, Matteo & Alessio; Nicola).

Awareness of the fact that properties obtained by an adjustment of thefigure ‘by eye’ do not grant the validity of the derived properties, leadspupils to look for a control, and they find it in an element obtained ‘byconstruction’, which for this very reason is reliable.

Strategy ‘by eye’ and strategy ‘by construction’ have definitely enteredthe practice and pupils use specific elements of the software (externalsigns) with the aim of keeping control of the two different meanings.

The previous example focusses on a basic aspect of the process of ex-ploration in the Cabri environment: the need to keep control of the figurein terms of given properties (hypothesis) and derived properties (thesis).The phenomena observable on the screen often hide the asymmetry of the


conditional relationship between properties, while in the dragging all theproperties hold at the same time.

Keeping control of the conditional by dragging is possible, but can bevery difficult: it requires that one

• relates the basic points (the only variable elements by dragging) andthe observed properties, and

• expresses such a relationship into a statement, connecting hypothesisand thesis.

It is interesting to remark that the pupils feel the need to control the givenproperties on the figure and to look for a support. The semantics drawnfrom the interaction with the software allows the pupils to generate ex-ternal signs to support the theoretical control which is not completelyinternalised.

7. CONCLUSIONS

This study, which is part of a collective research project on approachingtheoretical thinking at different age levels, shows interesting similaritieswith other experimental results (Mariotti et al., 1997; Bartolini Bussi etal., to appear).

In spite of the differences, it is interesting to remark the common fea-tures characterising and explaining the process of introduction to theoret-ical thinking as it is accomplished in different fields of experience.

Cultural artefacts, either microworlds or mechanical devices, may offersimilar support in the construction of meanings based on social interaction.

The field of experience of geometrical constructions in the Cabri en-vironment provides a context in which the development of the meaning ofGeometry theorem may be achieved.

The basic modification in which we are interested concerns the changeof the status of justification in geometrical problems. This modification isstrictly related to the passage from an ‘intuitive’ geometry as a collectionof evident properties to a ‘theoretical’ geometry, as a system of relationsamong statements, validated by proof. According to our basic hypothesis,the relation to geometrical knowledge is modified by the mediation offeredby the peculiar features of the software.

Mediation is a very common term in the literature concerning the useof computers in education. The term is not often explicitly defined, andsimply refers to the vague potentiality of fostering the relation betweenpupils and mathematical knowledge. A few authors directly discuss theidea of mediation and, among others, Hoyles and Noss do this in relation


to computers (Hoyles and Noss, 1996). The mediation function of the com-puter is related to the possibility of creating a channel of communicationbetween the teacher and the pupil based on a shared language (ibd. p.6).The potentialities of the software environments are related to the construc-tion of mathematical meanings that can be expressed by pupils throughinteraction with the computer.

Our analysis, which is based on the Vygotskian perspective and in par-ticular on the notion of semiotic mediation, has tried to outline and explainthe development of the meaning of proof.

The particular microworld offers a rich environment where differentelements may be used by the teacher as tools of semiotic mediation.

The basic element is the dragging function; at the very beginning drag-ging is an externally oriented tool, that introduces a perceptual test con-trolling the correctness of the solution to the construction task. As it be-comes part of interpersonal activities – mainly mathematical discussions– dragging changes its function and becomes a sign of the theoreticalcorrectness of the figure.

More generally, the Cabri environment offers a complex system of signssupporting the process of semiotic mediation, as it can be realised in thesocial activities of the class.

The fact that a command is activated by acting on a label, identifiedby the ‘name’ of a geometrical property, determines the use of a signfunctioning as control and as organiser of actions related to the task, i.e.construction procedures. Moreover, the fact that the commands availablemay be recognised as theoretical properties, corresponding to axioms ortheorems of a theory, makes the construction procedure itself an externalsign of a theorem. On the other hand the external sign (the word/commandor the construction) may function as internal control, as long as it refersto the possibility/necessity of explaining a theorem (its statement and itsproof).

The process of internalisation of such signs determines the construc-tion of the theoretical meaning of a construction problem and opens thetheoretical perspective for geometrical problems in general. The processof internalisation transforms the commands available in the Cabri menu –external signs – into internal psychological tools which control, organiseand direct pupils’ geometrical thinking, in producing both conjectures andproofs.

It is interesting to evidence the difficulty in the appropriation of thetheoretical control related to the dragging function. The process of inter-nalisation of the external control can be a slow process which must be


supported by the mediation of new external signs (offered by the software),that pupils autonomously create and use with that specific pupose.

The evolution of the pupils’ internal context is rooted in the construc-tion activities, where different processes such as conjecturing, arguing,proving, systematizing proofs as formal deduction are given sense andvalue. Our experiments also show that the contradiction highlighted byDuval between everyday argumentation and deductive reasoning, betweenempirical and geometrical knowledge, can be managed in dialectic termswithin the evolution of classroom culture.

In our experiment classroom culture is strongly determined by the re-course to mathematical discussion orchestrated by the teacher to changethe spontaneous attitude of students towards theoretical validation.

The introduction of computer environments in school mathematicalpractice has been debated for years, in particular the potentialities of Cabri,and dynamic geometry in general, have been described and discussed.With this experimental research study we hope to offer a contribution inillustrating the cognitive counterpart of classroom activities with Cabri thatallows an approach to theoretical thinking.

ACKNOWLEDGEMENTS

The research project has received financial support from C.N.R., M.U.R.S.T.(National Project ‘Ricerche di Matematica e Informatica per la Didattica’,coordinator: F. Arzarello). I wish to thank the teachers of the researchteam, M.P. Galli, D. Venturi e P. Nardini, who carried out the classroomexperiments and discussed with me the data on which this paper is based.I would also like to thank the many colleagues who in the past years havegenerously contributed to making me develop these ideas.

NOTES

1. The following remark concerns types of software which share with Cabri the generalfeature of a ‘drag mode’; I mean for instance Sketchpad or Geometric Supposer.

2. In this section I will employ the term ‘tools’ as used in the current English transla-tions of Vygotskij’s works. In current literature the terminology is rather confused, asdifferent authors use the same words with different meanings.

3. This unusual construction, quite different from those which can be found in the text-books, seems to provide a strong support to our interpretation.

4. After the construction of the angle bisector and the inclusion of the correspondingcommand in the available menu, the definition of perpendicularity was introduced.


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Boero, P., Dapueto, C., Ferrari, P., Ferrero, E., Garuti, R., Lemut, E., Parenti, L., Scali, E.:1995, ‘Aspects of the mathematics-culture relationship in mathematics teaching-learningin compulsory school’, Proc. of PME-XIX, Recife.

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Dipartimento di Matematica,Università di Pisa,Italy,E-mail: [email protected]

KEITH JONES

PROVIDING A FOUNDATION FOR DEDUCTIVE REASONING:STUDENTS’ INTERPRETATIONS WHEN USING DYNAMIC

GEOMETRY SOFTWARE AND THEIR EVOLVINGMATHEMATICAL EXPLANATIONS

ABSTRACT. A key issue for mathematics education is how children can be supported inshifting from ‘because it looks right’ or ‘because it works in these cases’ to convincingarguments which work in general. In geometry, forms of software usually known as dy-namic geometry environments may be useful as they can enable students to interact withgeometrical theory. Yet the meanings that students gain of deductive reasoning throughexperience with such software is likely to be shaped, not only by the tasks they tackleand their interactions with their teacher and with other students, but also by features ofthe software environment. In order to try to illuminate this latter phenomenon, and todetermine the longer-term influence of using such software, this paper reports on datafrom a longitudinal study of 12-year-old students’ interpretations of geometrical objectsand relationships when using dynamic geometry software. The focus of the paper is theprogressive mathematisation of the student’s sense of the software, examining their inter-pretations and using the explanations that students give of the geometrical properties ofvarious quadrilaterals that they construct as one indicator of this. The research suggeststhat the students’ explanations can evolve from imprecise, ‘everyday’ expressions, throughreasoning that is overtly mediated by the software environment, to mathematical explan-ations of the geometric situation that transcend the particular tool being used. This latterstage, it is suggested, should help to provide a foundation on which to build further notionsof deductive reasoning in mathematics.

KEY WORDS: appropriation of learning, computer environments, deductive reasoning,dynamic geometry software, geometry, mathematical explanation, mathematisation, medi-ation of learning, quadrilaterals, secondary school, sociocultural theory

INTRODUCTION

Providing a meaningful experience of deductive reasoning for students atthe school level appears to be difficult. A range of research has documentedthat even after considerable teaching input, many students fail to see a needfor deductive proving and/or are unable to distinguish between differentforms of mathematical reasoning such as explanation, argument, verifica-tion and proof (for recent reviews see Hanna and Jahnke, 1996; Dreyfus,1999). Amongst the reasons put forward for these student difficulties are


56 KEITH JONES

that learning to prove requires the co-ordination of a range of competen-cies each of which is, individually, far from trivial (Hoyles, 1997), thatteaching approaches often tend to concentrate on verification and devalueor omit exploration and explanation (de Villiers, 1998), and that learningto prove involves students making the difficult transition from a computa-tional view of mathematics to a view that conceives of mathematics as afield of intricately related structures (Dreyfus, 1999).

Despite the sheer complexity of learning to reason deductively in math-ematics, and the wealth of evidence suggesting how difficult it can befor students, there are studies that show that students can learn to arguemathematically. Research by, for instance, Maher and Martino (1996) andZack (1997) at the elementary school level, and by Dreyfus and Hadas(1996) and by Boero and colleagues (for example, Boero et al., 1996)at the high school level, illustrate how students can develop elements ofdeductive argument and how these notions of proving can depend on theclassroom ethos, the tasks the students tackle, and the form of interac-tions that take place between the students and between the teacher andthe students, as well as the tools available to the students. It is becauseof the complex nature of the interactions between these elements that, asDreyfus (1999) concludes, much remains unknown about how students’mathematical deductive reasoning evolves and changes.

In geometry, an area of the curriculum intimately connected with thedevelopment of the deductive method, computer software packages gener-ally known as dynamic geometry environments (DGEs) appear to havethe potential to provide students with direct experience of geometricaltheory and thereby break down what can be an unfortunate separationbetween geometrical construction and deduction (for a review see Chazanand Yerushalmy, 1998, p.72–77). As such, student use of a DGE couldhave an important role to play in enabling students to formulate deductiveexplanations and provide a foundation for developing ideas of proof andproving. Concerns remain, however, that the opportunity afforded by thesoftware of testing a myriad of diagrams through use of the ‘drag’ functionprovided by the DGE, or of confirming conjectures through measurements(that also adjust as the figure is dragged), may reduce the perceived needfor deductive proof (Chazan, 1993; Laborde, 1993; Hanna, 1998; Hoylesand Jones, 1998). If students are to gain facility with the deductive methodthrough experience with a DGE then a particularly important issue is theimpact that using such software has on the interpretation that students giveto the geometrical objects they encounter in this way and how they learn toexpress explanations and verifications of geometrical theorems, propertiesand classifications (see Laborde and Laborde, 1995; Hoyles 1995).

PROVIDING A FOUNDATION FOR DEDUCTIVE REASONING 57

Much previous research with dynamic geometry environments has quiteproperly focused on students in upper secondary (senior high) school wherethe students have received considerable teaching input in plane geometry,including the proving of elementary theorems, but may be new to theparticular software tool. What is less clear at the moment is what impactthe use of dynamic geometry software has on students in lower secondary(junior high) school where students have limited experience of the formalaspects of geometry but where contact with geometrical theory through thesoftware may be especially valuable in providing a foundation for furtherwork on developing deductive reasoning. At the moment too few studentssuccessfully make the transition to more formal mathematical study inwhich proof and proving are central. The aim of the study reported in thispaper is to contribute to what is known about enabling more students tosuccessfully make this transition.

In order to try to illuminate the impact of using such software, and todetermine the longer-term influence, this paper reports an analysis of datataken from a longitudinal study (see Jones, 1996, 1997, 1998) of lowersecondary (junior high) school students (aged 12 years old) learning as-pects of geometry in a particular DGE, in this case Cabri-géomètre version1.7 (Baulac et al., 1988). The focus of the paper is the progressive math-ematisation (see below) of the students’ sense of the software, reportingtheir interpretations and using explanations that the students give of thegeometrical properties of the figures they construct as one indicator of this.

In this paper, an explanation is taken to be “that which explains, makesclear or accounts for” (see the Oxford English Dictionary, 1989) and veri-fication is taken as “demonstration of the truth or correctness by fact orcircumstances” (op.cit.). In terms of mathematics, and following Balacheff(1988a: 2), a mathematical explanation is taken as “the discourse of anindividual intending to establish for somebody else the validity of a [math-ematical] statement”. In addition, and where these terms are used, a proofis “an explanation which is accepted by a community at a given time”,and a mathematical proof is “a proof accepted by mathematicians” (againfollowing Balacheff).

The analysis of the data from the longitudinal study that is reportedin this paper focuses on the students’ interpretations, especially the ap-propriation of mathematical terminology for explaining within geometriccontexts as this is mediated through the dynamic geometry environment.As Edwards (1997, p. 188) explains, “in order to effectively support theteaching of proof with meaning, we must understand how students learnto reason, how they come to perceive and describe mathematical patterns,how generalizations and mathematical arguments are constructed, and how

58 KEITH JONES

these processes can be supported in the learner” (emphasis in original).This paper concentrates on how the students reason about geometrical ob-jects and relations as they experience them through the dynamic geometryenvironment and how the mathematical explanations they offer evolve asthey become more experienced both with geometry and with the software.

STUDENT INTERPRETATIONS OF DYNAMIC GEOMETRY

ENVIRONMENTS

In this section a brief outline is given of some of the research findingsabout how students interpret geometrical objects and relations when usingdynamic geometry software. In general, as Goldenberg and Cuoco (1998)observe, as yet much remains unknown about how students glean geo-metric ideas from the complexly moving figures that they can encounterwith a DGE. What is known is that the computer environment affects theactions that are possible when solving problems (a task solved using dy-namic geometry software may require different strategies to the same tasksolved with paper and pencil) and it affects the feedback that is providedto the user (see Laborde, 1992, 1993a, 1993b). The DGE also introduces aspecific criterion of validation for the solution of a construction problem: asolution is valid if and only if it is not possible to “mess it up” by dragging(to use the expression adopted by Healy et al., 1994, see also Noss et al.,1994), or, in other words, that there is “robustness of a figure under drag”(as used by Balacheff and Sutherland, 1994, p. 147). This criterion of val-idation does not depend on the perceptive appearance of the product of theconstruction as this appearance can be modified using the drag facility. Topass this ‘drag test’ the figure has to be constructed in such a way that it isconsistent with geometrical theory.

Laborde (1993a, p.49) highlights an important distinction between draw-ing and figure: “drawing refers to the material entity while figure refers toa theoretical object”. In terms of a dynamic geometry package, a drawingcan be a juxtaposition of geometrical objects resembling closely the inten-ded construction (something that can be made to ‘look right’). In contrast, afigure additionally captures the relationships between the objects in such away that the figure is invariant when any basic object used in the construc-tion is dragged (in other words, that it passes the drag test). Hölzl (1995,1996) found that learners can get ‘stuck’ somewhere between a drawingand a figure. He suggests that this relates to the fact that in a DGE suchas Cabri-géomètre the verification process is controlled by the drag mode.He suggests that the more powerful the computer tool, the more didactic


efforts are needed to provoke pupils to focus on the relevant mathematicalrelationships.

A further open question at the moment is how students can distinguishfundamental characteristics of geometry from features that are the resultof the particular design of the DGE. As Balacheff (1996, p. 7) demon-strates with respect to Cabri-géomètre, the sequential organisation of ac-tions necessary to produce a figure in Cabri introduces an explicit orderof construction where, for most users, order is not normally expected ordoes not even matter. For example, Cabri-géomètre induces an orientationon objects: a segment AB is orientated because A is created before B.This probably makes sense in a direct manipulation environment, but it iscontradictory to the fact that in paper-and-pencil these objects have no ori-entation – unless it is explicitly stated. In a complex figure this sequentialorganisation produces what is, in effect, a hierarchy of dependencies aseach part of the construction depends on something created earlier. Hoyles(1995, p. 208) identifies this as a potential source of confusion in Cabri-géomètre as any hierarchy of relationships which has been establishedcannot then be modified (without undoing much that has been done, oreven starting the whole construction again).

When observing young students attempting to construct a rectangle us-ing the dynamic geometry software package Cabri, Hölzl et al. (1994, p.11) found that, in order to make any progress with such a task, the stu-dents had to come to terms with “the very essence of Cabri; that a figureconsists of relationships and that there is a hierarchy of dependencies”(emphasis in original). An example of this hierarchy of dependencies isthe difference (in Cabri1 for the PC) between basic point, point on objectand point of intersection. While all three types of point look identical onthe screen, basic points and points on objects are moveable (with obviousrestrictions on the latter). Yet a point of intersection cannot be dragged.This is because a point of intersection depends on the position of the basicobjects which intersect. Only the basic objects (such as basic points, lines,etc.) used in a construction can be dragged. Dependent objects, such aspoints of intersection, only move as a consequence of their dependency onthese basic objects. From their study, Hölzl et al. conclude that studentsneed to develop an awareness of such functional dependency if they are tobe successful with non-trivial geometrical construction tasks when usingdynamic geometry software. Such an idea as functional dependency is,within the dynamic geometry environment, intimately connected with thenotion of the robustness of a figure under drag mentioned above. Giventhe complexities involved, Hölzl et al. report that “not surprisingly, the

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idea of functional dependency has proved difficult [for students] to grasp”(op.cit.).

The above suggests that if DGEs are to be useful resources for helpingto build a foundation for deductive reasoning for younger students then it isimportant to know what interpretations these students make of geometricalobjects and relations experienced through a DGE. Of particular importanceis their sense of ‘drawing’ and ‘figure’ (following the work of Laborde andof Hölzl) and of the nature of geometrical objects and relations, especiallythe notion of dependency. The next section gives a concise outline of thetheoretical underpinning of the study reported in this paper.

THEORETICAL UNDERPINNING

In addition to a consideration of the previous research carried out withdynamic geometry software, some of which is referred to above, the theor-etical framework used to locate and inform the design, implementation andanalysis of the empirical component of the overall longitudinal study, fromwhich the data reported in this paper is taken, is derived from research inthe following areas:

a) Theoretical models of the teaching and learning of geometrical con-cepts, especially the van Hiele model (see, for example, van Hiele,1986; Fuys et al., 1988),

b) Theoretical perspectives on the teaching and learning of deductivereasoning, especially the work of Balacheff (1988a and b), de Villiers(1990, 1998), and Hanna (1990, 1998),

c) Sociocultural perspectives on learning, especially the work of Wertsch(1991, 1998),

d) Theoretical perspectives on the role of technological tools in the learn-ing process, especially the work of Pea (1987, 1993),

e) Theoretical perspectives on mathematisation, including the work ofGattegno (1988) and Wheeler (1982) together with, in a fairly restric-ted sense, aspects of mathematisation in work in the realistic math-ematics education (RME) paradigm, for example, Treffers (1987).

This paper is mostly concerned with the latter theoretical component, thenotion of mathematisation, which is dealt with in more detail below. Themain features of the other theoretical positions, as they relate to this par-ticular study, are as follows. The van Hiele model provides a backgroundframing for the study, especially as transitions between the van Hiele levelsare thought to be critical psychological shifts with important implicationsfor further learning. For example, in the van Hiele model the shift from


level 2 (identifying geometrical figures by their properties, which are seenas independent) to level 3 (recognising that a geometrical property of afigure precedes or follows from other properties), and subsequent progresswithin level 3, is especially important as these lay the foundations for fulldeductive proving in van Hiele level 4. The available evidence is that suchprogress made by school students is slow, with few students progressingbeyond level 2, even by the end of secondary (high) school (Senk, 1989).In terms of the use of dynamic geometry software, an in-depth case-studyof a single grade 6 student suggests that using such software may helpstudents to progress to the higher van Hiele levels (Choi koh, 1999). Whenexamining students’ deductive reasoning and proving processes, Balacheff(1988b: 216–218) found it useful to distinguish between what he called“pragmatic proofs”, which are “those having recourse to actual actions orshowings”, and “conceptual proofs”, which “do not involve action and reston formulations of the properties in question and relations between them”.In terms of what is learnt and how it is learnt, from the perspective ofsociocultural theory the assumption is that using a tool such as a dynamicgeometry package does not serve simply to facilitate mental processes thatwould otherwise exist. Instead, use of the software is thought to funda-mentally shape and transform the mental processes of the users (see Jones,1998; Mariotti and Bartolini Bussi, 1998; and, for a more general discus-sion of semiotic mediation, Bartolini Bussi and Mariotti, 1999). Finally,following from this and using the terminology of Pea (1987, 1993), explor-atory software environments in mathematics education, of which dynamicgeometry is an example, can be said to act as cognitive reorganisers ratherthan merely amplifiers of existing human capabilities.

As noted above, this paper mostly uses (a somewhat specialised versionof) the notion of mathematisation, and, indeed, progressive mathemat-isation (an idea taken from Treffers, 1987, see below). Mathematisation,following Gattegno (op.cit.) and Wheeler (op.cit.) involves a range of pro-cesses and facilities such as the ability to perceive relationships and toidealise them into purely mental material, the capacity to internalise ac-tions and such like (so as to ask “What would happen if?”), and the abilityto transform along a number of dimensions, such as from actions to percep-tions and from images to concepts. Most often it is used when looking at a‘real’ situation (as in the RME tradition), abstracting from it those elementsthat are wanted for closer study (or that appear to be tractable), setting amathematical model, making inferences within the model, checking to seewhether the results in the model are bourne out by observation and ex-periments, tinkering with the model to make it approximate reality better,and so on. Progressive mathematisation, as defined by Treffers (op.cit.),

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is the process whereby mathematical models are developed through thesuccessive positioning of contexts that embody the underlying structure ofthe concepts.

These ideas of mathematisation, and progressive mathematisation, areused in this paper in the following sense. In tackling tasks involving ele-mentary school geometry using dynamic geometry software, students canbe said to be involved in modelling the geometrical situation using the toolsavailable in the software. This involves setting up a construction and seeingif it is appropriate, and quite probably having to adjust the constructionto fit the specification of the problem (producing a ‘figure’, rather then a‘drawing’, in the sense of Laborde, above). For the purposes of this paper,this is taken as a form of mathematisation. When the tasks the students aretackling are linked in some way, such as involving the properties of quad-rilaterals (with a focus on the relationship between such properties), thismathematisation process becomes more like a process of progressive math-ematisation as the students develop a sense of the underlying relationshipsbetween the geometric properties.

In the next section the precise methodology is described. Overall, theresearch design follows the example of Meira in focusing on how “in-structional artifacts and representational systems are actually used andtransformed by students in activity” (Meira, 1995, p. 103, emphasis inoriginal) rather than solely asking whether the students learn particularaspects of geometry better by using a tool such as a DGE when comparedto using other tools (such as ruler and compass). The reason for this is thatthe focus of interest is both what the students learn and how they learn it.

RESEARCH DESIGN

The empirical work for this study was designed to be carried out in theUK and, following Hoyles (1997), the design was informed by the struc-ture of the mathematics curriculum experienced by students in the UK.As Hoyles describes, while formal proof is likely to be restricted in theUK to the most able students only (and probably only encountered bystudents in upper secondary school), the curriculum does provide for op-portunities for conjecturing and presenting generalisations at all levels.In terms of geometry, the curriculum attempts to incorporate aspects ofthe following geometries: plane, analytic/co-ordinate, and transformation.Such curriculum considerations mean that students in lower secondary(junior high) school typically know, for example, some of the propertiesof certain geometrical figures, have some experience of conjecturing anddescribing observations in open-ended problem situations, but have not


been introduced to the formal aspects of proof and proving. In terms ofexperience with computer environments, students typically have followedan information technology course, which means they have some facilitywith using the computer mouse, menu items, and computer files.

The geometrical topic chosen as most suitable for the empirical studywas the classifying of quadrilaterals. There were two main reasons for thisdecision. First, in terms of the ‘family of quadrilaterals’, there has beenboth discussion about the classification of quadrilaterals (for example, deVilliers 1994), and research involving students’ ability to classify them,including both research using the van Hiele model (for instance, Fuys etal., 1988), and research involving pupils using the mathematical program-ming environment Logo (see Hoyles and Noss, 1992, for a comprehensivereview). As de Villiers (1994, p. 11–12) explains, classifying is closelyrelated to defining (and vice versa) and classifications can be hierarch-ical (by using inclusive definitions, such as a trapezium or trapezoid is aquadrilateral with at least one pair of sides parallel – which means thata parallelogram is a special form of trapezium) or partitional (by usingexclusive definitions, such as a trapezium is a quadrilateral with only onepair of sides parallel, which excludes parallelograms from being classi-fied as a special form of trapezium). In general, in mathematics, inclusivedefinitions (and thus hierarchical classifications) are preferred (although itshould be stressed that exclusive definitions and partitional classificationsare certainly not incorrect mathematically, just less useful). De Villiers(1994, p. 17) quotes from a number of studies (including Fuys et al.,1988) that have shown very clearly that many students have problems withthe hierarchical classification of quadrilaterals. He suggests that some ofthe difficulties “do not necessarily lie with the logic of inclusion as such,but often with the meaning of the activity, both linguistic and functional:linguistic in the sense of correctly interpreting the language used for classinclusions, and functional in the sense of understanding why it is moreuseful than a partition classification”. He observes that dynamic geometrysoftware “offers great potential for conceptually enabling many childrento see and accept the possibility of hierarchical inclusions (for example,. . . letting them drag the vertices of a dynamic parallelogram . . . to trans-form it into a rectangle, rhombus or square)”. These comments from deVilliers, together with a reading of the other work on the classification ofquadrilaterals, informed the precise form of the empirical work describedbelow.

The second reason for choosing the classifying of quadrilaterals wasmore practical than epistemological. It was that one major strand within thegeometry component of the UK mathematics curriculum is understanding

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and using the properties of plane shapes. In the lower secondary years(ages 11–13), the emphasis on this component of the curriculum movesfrom (informally) recognising and sorting geometrical figures towards theformal definitions required to classify and deduce properties of, and rela-tions between, such figures. Choosing this topic meant that it fitted with theschool’s programme of study and no special dispensation would be neededto incorporate the topic (in the UK the curriculum is statutory and specialdispensation is required if a school wishes to vary from it). Choosing acommonly occurring topic also means that the research findings may befound useful by practising secondary mathematics teachers.

Wherever possible, design choices were made with a view to the typ-icality of the setting. The school selected for the empirical work was anurban comprehensive school whose results in mathematics at age 16 wereat the national average (there is a national system of testing in the UKthat allows such judgements to be made). The mathematics teachers inthe school used a problem-based approach to teaching mathematics andthe students usually worked in pairs or small groups on mathematicalproblems and occasionally used computers. Throughout their mathematicswork the students were expected to be able to explain the mathematics theywere doing, either orally or in writing. This meant that work on geometryusing dynamic geometry software would fit with the usual experience ofthe students. The classes of 12 year-olds in the school had four 50-minutemathematics lessons per week.

The particular class of 12 year-olds selected for the research were judgedtypical of that suitable for studying the relationships between quadrilater-als in that they were above-average in mathematics for their age (the schoolallocated students to different mathematics classes according to attainmentin mathematics tests). A teaching unit was developed in collaboration withthe teacher of the class that would address the properties of quadrilateralsand could be accommodated in the regular routine of the class. For mostof the 9 months of the study up to four computers were available in theclassroom. This meant that, as pairs of students took turns in using thecomputers, there might be gaps of up to a week between sessions thatany particular pair of students had using the software. During these ‘gaps’the students undertook other mathematics work, including some geometrytopics involving area and volume, but not directly about the geometricproperties of quadrilaterals. The version of Cabri-géomètre in use wasCabri I for the PC.

All the students in the class were tested using a van Hiele test (Usiskin,1982) at the start of the unit of work and on its completion. The teachingunit was prepared to form three phases, and designed to fit around other


Figure 1. An example task from phase 1.

mathematics work for the class. During each of the phases, the studentsworked in pairs (usually the pairs they worked in for all their mathematicswork).

During phase 1 the students gained preliminary experience with Cabri-géomètre while working through a short series of tasks involving lines andcircles (see Figure 1 for an example). The aim of the phase was for stu-dents to acquire familiarity with the software interface and be introducedto the constraint of robustness of a figure under drag (see Balacheff andSutherland, 1994, p. 147). For each task (in phase 1 and in the subsequentphases) the challenge for the students was to reproduce, using the software,a figure identical to one provided on paper but which could not be ‘messedup’ (this phrase suggested by Healy et al., 1994, was used consistentlywith the students). Figure 1 shows an example of one of the tasks fromphase 1. The tasks used in each of the three phases of the study wereso designed that to successfully meet the challenge of constructing thefigures so that they are invariant under drag, the students have to analysethe spatial arrangements (taken as a form of mathematisation) and, as theyare novices with the software, work out how to realise their constructionsin the software environment in such a way that not only does it appear to becorrect visually in a static form but that if any objects (such as points, linesor circles) used in the construction are dragged, the patterns remain con-sistent. Following Laborde (1993a, p. 49), the challenge for the students isto produce a ‘figure’ (which makes use of geometrical relationships) ratherthen a ‘drawing’ (which only looks like the required figure and which failsthe dragging test of validity). For most students, phase 1 took up to threehours of using the software.

Phase 2 of the teaching unit involved the students working through aseries of three tasks that required constructing the following quadrilaterals:a rhombus, a square, and a kite. Each task contained a visual prompt andthe challenge to construct the figure so that it was invariant under drag andexplain why the figure constructed is a particular quadrilateral. Figure 3

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Figure 2. Visual prompt for constructing a square (task 2, phase 2).

shows the visual prompt the students received for constructing a square(task 2 in the phase 2 sequence of tasks).

In consonance with the idea of progressive mathematisation, the tasksin phase 2 were designed with the intention that the students would becomemore adept at analysing the geometrical structure provided in the visualprompt and, by having done this, be able to use this analysis to explainwhy the shape was the particular quadrilateral. The sequencing of the tasks(rhombus, square, kite) was designed with the intention that students mightuse approaches and geometrical properties gleaned from solving earliertasks in approaching the later ones (including those in phase 3 below).While, initially, the notion of robustness under drag was introduced as achallenge to the students, both phase 2 of the teaching unit and phase 3were designed so that this notion became intimately connected with thegeometrical properties of the quadrilaterals being constructed. In this wayit was intended that the students would come to appreciate, in terms ofgeometrical reasoning, the significance of robustness under drag and itsusefulness as a test of the validity of constructions in terms of geometricaltheory. The students took about two hours to complete phase 2 of theteaching unit.

Phase 3 of the teaching unit involved the students working through aseries of six tasks that involved relationships between various quadrilater-als: the rhombus and the square, the rectangle and the square, the kite and


the rhombus, the parallelogram and the trapezium, the rhombus, rectangleand the parallelogram. Most of the students took up to three hours onthis phase of the teaching unit. For each task, the students were providedwith a visual prompt similar to Figure 2 and the challenge to constructthe figure so that it was invariant under drag and explain why all squaresare rectangles, for example, or why all rectangles are parallelograms. Thereason for this choice of phrase is to investigate the students’ developingconception of this inclusive mathematical classification.

The sixth and final task of phase 3 asked the students to complete ahierarchical (inclusive) classification of the ‘family’ of quadrilaterals andexplaining the relationships within this ‘family’ (see Figures 3a and 3b).In this final task the phrasing used was that a particular quadrilateral (say asquare) was “a special case” of another quadrilateral (in this example, bothof a rhombus and a rectangle). This phrasing was chosen as another way ofexpressing inclusive mathematical classification. Through the use of thesedifferent phrasings (that all squares are rectangles, and that a square isa special case of a rectangle) an attempt was made to take account ofwhat Hershkowitz (1990, p. 81) calls “the opposing direction inclusionrelationship” between sets and subsets of examples of, say, quadrilaterals,on the one hand, and the sets and subsets of their attributes on the other(for example, that the set of squares is included in the set of parallelograms,which, in turn, is included in the set of quadrilaterals, but the set of criticalattributes of squares includes the set of critical attributes of parallelograms,which includes the set of critical attributes of quadrilaterals). The impact ofthis opposing direction inclusion relationship is that, for example, youngchildren may not entertain a square as a quadrilateral because a square hasfour equal sides while other quadrilaterals do not.

During each phase, efforts were made to keep interventions by theteacher and the researcher to the following: responses (often in the formof questions) to student questions, and asking students for explanations.At times, suggestions were offered when students did not know how toproceed. The timing and nature of these occasions were noted. Particularattention was paid to instances when technical geometric terminology wasintroduced and used. The nature and impact of the interventions is animportant aspect of the analysis of the data from this study and will bereported elsewhere. In planning the empirical study it was anticipated thatthe nature of the interventions would change during the various phasesof the teaching unit, from ones concerned with aspects of the software inphase 1, to ones relating the properties of particular quadrilaterals in phase2, and ones involving the relationships between quadrilaterals in phase 3. Aconscious decision was made to focus the interventions on the geometrical

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properties of the quadrilaterals and not to encourage use of the measuringtools (such as length and angle) available in the software environment.Following the theoretical framework and the usual practice of the classteacher, the nature of the interventions was designed to be consonant withguided participation in sociocultural activity. As noted above, the norm ofthe classroom was for students to provide explanations and for these to berelated to the structure of the problems being tackled.

There were 28 students in the experimental class. Seven pairs of stu-dents were studied during phase 1 in order to select four pairs for detailedstudy during phases 2 and 3. The four pairs of students studied duringphases 2 and 3 were selected both to represent the range of attainment inthe class in terms of the van Hiele levels and on the basis that each pairworked reasonably well together. The four pairs were as follows:

– Pair A: both students van Hiele level 1–2– Pair B: one student van Hiele level 1–2, the other level 2– Pair C: both students van Hiele level 2– Pair D: one student van Hiele level 2, the other level 2–3

For all the pairs of students studied, the following data was collected:video and additional audio tape to capture the onscreen work and student-student and student-teacher interactions, student written work (unaided),student software files, the ‘history’ of the student constructions using thesoftware (a feature available with the particular software), and researcherfield-notes.

RESULTS AND ANALYSIS

The analysis presented below focuses on data from two pairs of students,pairs A and C, chosen because the individuals in each pair were assessed asbeing at similar van Hiele levels. According to the van Hiele model, as vanHiele explains, “each [van Hiele] level has its own linguistic symbols andits own relations connecting these symbols. A relation which is ‘correct’at one level can reveal itself to be incorrect at another. Think for example,of a relation between a square and a rectangle. Two people who reasonat different levels cannot understand each other. Neither can manage tofollow the thought processes of the other” (van Hiele, 1959; as quotedin Fuys et al., 1988, p. 6). Thus choosing pairs A and C in this papermeans that the analysis can focus more precisely on the interpretations andexplanations of the students without being over-complicated by possiblemisunderstandings between the students. In the overall research design,pairs B and D were chosen in order to examine these inter-student aspects


in more detail, particularly when one student is more capable (in terms ofthe van Hiele levels) than the other. These inter-student aspects are subjectto separate analysis. Suffice to say for this paper that the overall pattern ofdevelopment analysed below is typical of that found with all the pairs ofstudents.

In the analysis the term ‘mathematical reasoning’ is used to denotelogical inference and deduction of a form appropriate to lower second-ary (junior high) school students (and should not be taken to mean theuse of abstract symbolic notation, truth tables or formal axiomatic proofs,for example). In the context of this study mathematical reasoning can in-volve using the properties of shapes to judge the validity of results andjustifying steps in giving explanations for statements. Precision in math-ematical reasoning is taken to include the use of the mathematical termsthat are appropriate for lower secondary school students, as opposed to‘everyday’ terms for geometrical objects such as ‘oblong’, ‘diamond’ or‘oval’. Thus ‘mathematical reasoning’ is taken to mean making reasonablyprecise statements and deductions about properties and relationships. Suchreasoning can be quite detailed without necessarily being thorough enoughto be called a proof.

The first part of this section reports on phase 1 of the teaching unit,focusing in particular on how the students interpreted the notion of theconstraint of robustness of a figure under drag. The second part of thesection reports on the two pairs of students as they work through a seriesof tasks that involved constructing various quadrilaterals. The focus is theirinterpretations using their evolving mathematical explanations as an indic-ation of this as they attempt to explain the properties of the quadrilateralsthey construct.

Phase 1 of the teaching unit

Phase 1 of the teaching unit served to introduce the students to the softwareand to the constraint of robustness of a figure under drag. In this phasethe students worked through a short series of tasks involving patterns ofinterlinking lines and circles (see Figure 1 for an example).

The data from this phase raises several issues about the interpretationthe students made of the software environment. Some of these were fairlytrivial. For example, the distinction between lines (that are infinite) andline segments – a distinction that these particular students had not metpreviously. Other aspects of the software environment took longer for thestudents to become accustomed to. The most important were:

1. The aspect of functional dependency as realised in the software wherebysome objects can be dragged while others cannot.

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2. The distinctive properties of points of intersection (which, in Cabri 1,had to be explicitly created).

3. The sequential organisation required in order to construct a figure thatis robust under drag.

These three aspects relate to phenomena created by the software (and onlyrelate to geometrical theory in terms of how this theory is realised in thesoftware) and it is crucial that students recognise this if they are going tosee through the software to the geometry.

For the students in the study reported in this paper, the notion of theconstraint of robustness of a figure under drag became linked with usingpoints of intersection to try to hold the figure together. To illustrate this,the extracts below are from the transcribed sessions and written work ofpair C (pseudonyms Heather and Karol). In all the data extracts presentedbelow, square brackets are used to insert short phrases in order to clarifythe meaning of the extracts. Normal use is made of question marks andexclamation marks. Short pauses are shown by a series of dots.

On two occasions during session 1 (of phase 1), the students raisedquestions and received input from the teacher about dependency and aboutpoints of intersection. On the first occasion the students want to delete apoint. When attempting to do so they get the following message from thesoftware: ‘Delete this object and its dependents?’. They ask the teacherwhat this means. The teacher suggests that they go ahead and delete thepoint and see what happens (after reassuring them that they can undo thedelete). The students delete the point and two line segments disappear. Thisgives the teacher an opportunity to explicitly refer to dependency:

Teacher: that bit of line depended on that point, and that bit of line did, so theyboth went.

[Pair C, session 1 (phase 1)]

Later in the session the students discover that the points of intersectionthey have constructed cannot be dragged. Here the teacher again refers todependency and explains how points of intersection depend on the otherobjects. After a little thought and dragging, one of the students says:

Karol: You can’t drag that point [a point of intersection] because it is dependenton them [indicating the points used to create the shape].


The other student nods, which appears to indicate that, at this point, the stu-dents appreciate what is different about points of intersection and how this


relates to dependency. During the next session, however, there is evidencethat they have developed their own, somewhat different, interpretation.

In session 2, while tackling another task involving lines and circles, thefollowing exchange takes place:

Karol: What’s the [point of] intersection doing? Does it keep the dot [the point]there?

Teacher: What you are finding is the point here, where the circle crosses the line.Karol: Right, so if it was like that [indicating a different arrangement of lines],

then it [the point of intersection] would be there.Teacher: It is always where the lines cross.


In raising the question about points of intersection, here is a first indicationof the interpretation of the students have of points of intersection. It seemsthat Karol may think that such points have a role in ‘holding’ a figuretogether so that it is invariant under drag. Later in the session the otherstudent gives another indication:

Heather: You have to make an intersection between those two lines so that theycan’t be moved.


In this case it is not altogether clear what the student means. However, thestudents do successfully complete the tasks for the session and, at the endof the session, are asked why their figures cannot be ‘messed up’. Heatherreplies:

Heather: They stay together because of the intersections.


This statement is not incorrect as it relates to their perspective on robust-ness under drag, but it also masks another interpretation that is revealedin the next session. In their third session, the students are in the processof constructing a rhombus, which they need to ensure is invariant whenany basic point used in its construction is dragged. As they go about con-structing a number of points of intersection, one of the students commentsspontaneously:

Heather: [referring to a point of intersection] a bit like glue really. It just gluedthem together.

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This spontaneous use of the term ‘glue’ to refer to points of intersectionhas been observed by other researchers (see, for example, Pratt and Ainley,1996) and is all the more striking given the fact that earlier on in the lessonthe students had confidently referred to such points as points of intersection(implying that such points only exist if other geometrical objects intersect).

A little later in the same session, one student in the pair again asks theteacher why it is not possible to ‘drag’ points of intersection. The teacherreplies:

Teacher: Because the intersection points just show you where two things cross.Karol: So how come it keeps it together if it’s just a dot to show you where they

cross?Teacher: You can move that point because it’s the centre of the first circle that

you drew. So if you move that [point], then because you are changing thesize of the first circle, the point where it crosses the other circle changes sothat changes the other circle.

Karol: So that changes everything.Teacher: Because the other circle depends on that.Heather: So because it depends on it, it moves.


Here one of the students, Karol, asks, “So how come it [a point of intersec-tion] keeps it [the figure] together. . .?”. The teacher does not address thisdirectly but returns to the idea of dependency.

At the end of each session the students are expected to write downsomething about the session. Below is an extract from what this pair writesat the end of the following session:

Things are ‘dependent’ and when they are made dependent they can’t move.Dependence is created by an intersection between two things. For this ex-ercise we made everything dependent on the perpendicular line so althoughthings move in different ways, one thing holds everything in place.You have to make them dependent on each other, or another object, so itstays how we want it to.


In writing that things “can’t move” the student is employing their owndescription of invariance under drag. Clearly, under drag, things do move,including points of intersection. The ‘things’ that do not ‘move’ are thegeometrical relationships that have been constructed (parallel lines remain


parallel, for example). It is the hierarchies of dependencies, and the se-quential organisation required by the software (see Balacheff, 1996, p. 7),that the students are beginning to appreciate are central to constructingfigures in a DGE (such that any particular construction is robust underdrag). These are features specific to the software and the way that geomet-rical theory is realised in that environment. As noted above, Hölzl et al.(1994) found that students in their study needed to develop an awareness ofsuch functional dependency if they were to be successful with non-trivialgeometrical construction tasks using a DGE. Their experience was that thisidea was difficult for students to grasp. The students in this study similarlytook time to grasp the idea of functional dependency. As also noted above,the spontaneous use of the conception of points of intersection as ‘glue’,see student comment from session 3 (phase 2), is a phenomenon observedby other studies of younger students’ early experiences with Cabri (see, forexample, Pratt and Ainley, 1996). It occurred spontaneously in this studyreported, despite the planned interventions by the teacher that focused onthe notion of dependency. Further evidence of the difficulty students havewith interpreting points of intersection is provided by Hoyles (1995, pp.210–211).

The above examples illustrate how the notion of the constraint of ro-bustness of a figure under drag became linked with using points of inter-section to try to hold the figure together. The focus for the students wason the mechanical aspects of the software environment, as illustrated bytheir attempts to use points of intersection to ensure that their constructioncould not be ‘messed up’, rather than on the geometry of the figure beingconstructed. These interpretations of the software environment made bythe students in phase 1 (and into phase 2) of the teaching unit is an import-ant aspect of the progressive mathematisation of the students’ sense of thesoftware environment. It provides part of the basis by which the studentsbegan tackling the tasks in phases 2 and 3, which involve geometricaltheory much more explicitly.

In the analysis below, of phases 2 and 3, a major source of data is theunaided writing of the student pairs that they produced during and at theend of each session. This is augmented by extracts from the transcribedrecordings of the student oral explanations. The reason for focusing onthe student explanations is to reveal how they progressively mathematisethe sense they made of the software environment and how this impacts ontheir developing mathematical reasoning. Such a focus means, however,that, in general, there is little space to show how the students came to theirparticular explanations. This is not the focus of this paper. Data from theoverall corpus is selected for each of phases 2 and 3 to illustrate the main

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aspects. The selection has been made on the basis of the representativenessof the data in illustrating the overall trend of the students’ progressivemathematisation.


Task 1 of phase 2 asked the students to construct a rhombus and explainwhy the shape is a rhombus (the task is similar in format to that for con-structing a square as shown in Figure 2). The explanations of the studentsare as follows:

Pair A written explanation:

The radius is the same for the circle and the diamond [the rhombus] and wemade the diamond from the help of the first construction. The sides are allthe same because if the centre is in the right place the sides are bound tobe the same. The diagonals of the diamond cross in the middle though theyare different size (length). They cross at the middle through the line. Theirdiagonals bisect each other. The angles [at the intersection of the diagonals]are all the same. They are 90◦. The opposite angles [of the rhombus] are thesame. Two are more than 90◦ but less than 180◦ and the others are less than90◦ but more than 0◦.This shape is a rhombus because the sides are the same, the diagonals bisectat right angles and the opposites have the same angles.

Pair C, in their written explanation, concentrate on recording their under-standing of dependency (see above). Below is an extract from the sessiontranscript that took place near the end of the session:

Teacher: What sort of shape is that?Karol: It’s a rhombus.Teacher: How do you know it’s a rhombus?Karol: Our old maths teacher used to call a rhombus a drunken square, because

it’s like a square, only sick.Teacher: What do you know about a rhombus, from what you have done?Heather: It’s got a centre.Karol: It’s like a diamond . . .. But it’s not a square.Teacher: What can you say about the sides or the angles . . . or the diagonals?Karol: Those two angles [indicating the angles at one pair of opposite vertices]

are the same, and those two are the same . . . [indicating the other pair ofopposite angles]But they are not all the same [indicating adjacent angles]And . . . the sides are all the same length . . . I think.

Heather: It’s the same distance across each side.Teacher: What can you say about how the diagonals cross?


Karol: A right angle.Teacher: How do you know?Karol: It looks straight.

These outcomes are essentially descriptive rather than explanatory, andthere is evidence of a lack of student capability with precise mathematicalterminology (viz. the use of ‘diamond’ by both pairs) and some relianceon perception rather than mathematical reasoning (for example, “It looksstraight”). As the students are familiar with the properties of a rhombusthey are able to provide these. However, they are likely to be unfamiliarwith the notion of economic mathematical definitions (de Villiers, 1994, p.12), that is, definitions that contain only necessary and sufficient proper-ties, and with how the properties are related (beyond writing, for example,that “if the centre is in the right place the sides are bound to be the same”).

In task 2 of phase 2, the students are asked to construct a square andexplain why the shape is a square (see Figure 2). The written explanationsof the students are as follows:


It is a square because the sides are equal and the diagonals intersect. Thediagonals are [at] right angles (90◦).

Pair C written explanation:

It [the square] is made up of four equal sides. Its diagonals are equal. Weknow the diagonals are equal because they are the diameters of the circle.The diagonals cross at 90 degrees. The diameters have to be equal for it tobe a circle, and [the] diagonals have to be equal at 90 degrees [for it] to bea square.It is a square because two equal diagonals cross each other at a 90 degreeangle.

For this task, the written work of pair A remains essentially descriptivewhile pair C (the more able pair) include an explanation that the diagonalsof the square are equal “because they are the diameters of the circle”. Forboth pairs, their use of mathematical terminology is more precise than intask 1.


While the three tasks in phase 2 (on which data from two is providedabove) were concerned with constructing individual quadrilaterals, the tasksin phase 3 involved constructing a specified quadrilateral (for example, arectangle) in such a way that by dragging one of the vertices it could bemodified (or transformed) into a special case (in the example of a rectangle,the special case would be a square).

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Task 5, of the overall sequence of tasks on quadrilaterals (the first taskin phase 3), required the construction of a rectangle that could be modifiedto a square. The students were then expected to explain why all squares arerectangles. These are their written explanations:


A rectangle . . . becomes a square when the diagonals become right angleswhere they meet.


You can make a rectangle into a square by dragging one side shorter and sothe others become longer until the sides become equal.

These are very reasonable explanations of how a rectangle can “become asquare”. As can be seen from the students’ explanations, these are couchedin terms of the nature of the software environment, pair C more overtly(through use of the term “dragging”). Both pairs write about something be-coming something. These ‘pragmatic’ explanations, it is proposed, are ana-logous to the notion of ‘pragmatic’ proofs, being ones that have “recourseto actual actions” (Balacheff, 1988b, pp. 216–218).

What is not evidenced here, at this point, is whether the students ap-preciate that, using inclusive definitions, a square is a special case of arectangle (which they may feel is different from a rectangle being ableto “become” a square) or, similarly, that all squares are rectangles (eventhough they were asked about that). Partly, of course, this is the result ofthe task, yet the task depends on the software. In the DGE, by definition, aquadrilateral constructed as a square cannot be modified to a non-square asthe drag test of validity means that it must remain a square whatever basicobjects used in its construction are dragged.

Task 6 required the students to construct a kite that could be modifiedto a rhombus and explain why all rhombi are kites. In task 7 the studentsare asked to construct a trapezium that can be modified to a parallelogramand thereby explain why all parallelograms are trapeziums. These are theirwritten explanations:


It is a trapezium because it has one pair of parallel lines. A parallelogram isparallel both ways.


Trapeziums have one set of parallel lines and parallelograms have two setsof parallel lines.

Neither of the students’ written explanations is couched in terms of thenature of the software environment. There appears to have been a shift


from ‘pragmatic’ explanations that reflect the nature of the software en-vironment to mathematical explanation. The cause of this shift is mainlydue to the role of the teacher in consistently referring to the geometricalproperties of the shapes whenever there was an interaction with the stu-dents. The role played by the teacher and the impact of the teacher-studentinteractions is the subject of separate analysis.

Task 9 asked the students to complete the worksheet shown in Fig-ure 3a as a way of showing the relationships between the ‘family’ ofquadrilaterals

All the pairs of students completed this task satisfactorily, in each casewith some interaction with the teacher. This interaction consisted of theteacher referring the students back to earlier tasks they had completed inthe teaching unit, and the teacher asking the students to explain any rela-tionship between the various quadrilaterals that the students could identify.A completed worksheet (from pair A) is given in Figure 3b. Below areextracts from the transcripts of the lessons relating to this task involvingpairs A and C.

Extracts from Pair A session transcript (pseudonyms Harri and Rus-sell):

Teacher: Why is a square a special sort of rectangle?Russell: Because they’ve both got right angles [at the vertices] but with a rect-

angle [indicating one that is not a square] one of the sides is bigger than theother.

Teacher: Why is a rectangle a special case of a parallelogram?Harri: The two opposite [sides] are the same length but [indicating a parallelo-

gram that is not a rectangle] they [the angles at the vertices] are not rightangles.

Extracts from Pair C session transcript (pseudonyms Heather and Karol):

Teacher: Why is a square a special sort of rhombus?Heather: Because in a square all the . . . all the corners are 90 degrees and all the

sides are equal, but in a rhombus [indicating one that is not a square] all thesides are equal but they [the angles at the vertices] are not 90-degree angles.

Teacher: Why that arrow? [indicating that a rhombus is a special form of paral-lelogram].

Karol: It’s just like the rhombus and the square because . . . because all the sides. . . the sides are . . . the opposite sides are of equal length, but there [in the

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Figure 3a. Worksheet on the ‘family’ of quadrilaterals.

Figure 3b. Worksheet on the ‘family’ of quadrilaterals.


rhombus] they [the diagonals] cross at 90 degrees and there [indicating aparallelogram that is not a rhombus] they don’t.

Thus, by the end of the teaching unit, the students were able to acceptquestions of the form ‘why is one quadrilateral a special case of another’and could offer explanations as to why this is the case. These explanationsare interesting in terms of how difficult it was for the students to articulatetheir explanations without recourse to concrete illustrations. Consider theexplanation given by Russell in response to the question, why is a square aspecial sort of rectangle. Saying that “they’ve both got right angles [at thevertices] but with a rectangle [indicating one that is not a square] one ofthe sides is bigger than the other” could be taken as incorrect, mathematic-ally, because, as the set of rectangles contains the set of squares (using aninclusive classification), it is incorrect to say that a rectangle “has one ofthe sides bigger than the other” (since the statement has to refer to squarestoo, for which it is patently untrue). The same argument can be made aboutthe explanations offered by the other students. Each of them had to find away of indicating that the set of more general quadrilaterals to which theywere referring excluded the special case. This is because, as de Villiers(1994: 13) explains, partitioning is a “spontaneous and natural strategy”and that “we would normally call a square a ‘square’ and reserve the term‘rectangle’ only for a non-square (or general) rectangle”.

In explaining why a square is a special sort of rectangle, Russell couldhave made use of a term such as ‘oblong’ for those rectangles that are notsquares. Yet this term is not only superfluous in an inclusive classificationbut it depends itself on an exclusive definition, something that it not alwayshelpful from a mathematical perspective and is often discouraged in math-ematics curricula and textbooks. The alternative would have been for thestudent to have used a much more complicated sentence structure, some-thing like, “both squares and rectangles have right angles at their verticesbut rectangles that are not squares have one of the sides bigger than theother”. This is where issues to do with the logic of inclusive definitions andclassification, the language used for class inclusions, and the functionalaspects of hierarchical classification all have a bearing.

DISCUSSION

The classroom used for this study was selected because the students wereaccustomed to a pedagogical approach involving mathematisation. Theusual approach of the teacher was to have the students working in pairsor small groups (of 3 to 6 students) on a range of problem-based tasks,

80 KEITH JONES

some ‘real’ (and similar to those in the RME tradition) and some of amore ‘pure’ mathematical nature. The tasks the students tackled in theteaching unit developed for this study was of the latter form. Their pro-gressive mathematisation over the period of the study can be summarisedas follows:

• Initially, an emphasis on description rather than explanation. Somereliance on perception rather than mathematical reasoning. Lack ofcapability with precise mathematical language (similar to that foundin other studies, for example Fuys et al., 1988, pp. 135–136).

• At an interim stage, explanations become more mathematically pre-cise but are influenced (mediated) by the nature of the dynamic geo-metry software (for example by the use of the term ‘dragging’ or byother phrases linked to the dynamic nature of the software).

• At the end of the teaching unit, explanations related entirely to themathematical context.

Overall, as the students worked through the teaching unit, there was a shiftin their thinking from imprecise, ‘everyday’ expressions, through reason-ing mediated by the software environment to mathematical explanations ofthe geometric situation.

Given the significant problems that many students have with the hier-archical classification of quadrilaterals (see, for example, Fuys et al., 1988and de Villiers, 1994) the evidence reported in this paper supports the sug-gestion by de Villiers (op.cit.: p. 17) that dynamic geometry software offers“great potential for conceptually enabling many children to see and acceptthe possibility of hierarchical inclusions”. Such an outcome should helpto lay a solid foundation on which to develop further notions of deductivethinking.

The research study reported in this paper also reveals the mediationalimpact of using dynamic geometry software. As documented by this study(and by other research referred to in this paper), this mediational impactwas in terms of the following:

• The students’ understanding that the order in which objects werecreated leads to a hierarchy of functional dependency within a figure.

• The constraint of robustness of a figure under drag becoming linkedwith using points of intersection to try to hold the figure together.

• The ‘dynamic’ nature of the software influencing the form of explan-ation given by the students.

Thus, when using dynamic geometry software, students need to come toterms with the notion of a hierarchy of functional dependency within afigure (see, for further examples, Hölzl et al., 1994; Jones, 1996). Secondly,


the students need to gain an appreciation of the notion of the constraint ofrobustness of a figure under drag as a mathematical feature, rather than,say, as ‘mechanical glue’ (Pratt and Ainley, 1996; Jones, 1998). Thirdly,the ‘dynamic’ nature of the software influences the form of explanationgiven by the students (what Hölzl, 1996, p. 184, refers to as reasoning “ina Cabri-specific style”).

As Hölzl (1996) points out, this latter example of tool mediation is akinto the notion of ‘situated abstraction’ proposed by Noss and Hoyles (1996,pp. 122–125) as a step in constructing a mathematical generalisation. Inthis conceptualisation, the abstraction is ‘situated’ in that the knowledge isdefined by the actions within a context. Yet it is an abstraction in that thedescription is not a routinised report of action but exemplifies the students’reflections on their actions as they strive to communicate a mathematicalexplanation. Recall that Pair C wrote: “you can make a rectangle into asquare by dragging one side shorter and so the others become longer untilthe sides become equal”. Such an explanation, using Balacheff’s (1988)distinction, could be called a pragmatic explanation in that it refers toactual actions. The form of explanations given by the students later in theteaching unit, in resting on the mathematical properties in question, would,to continue the analogy, constitute conceptual explanations.

The evidence from this study indicates that using dynamic geometrysoftware does provide students with access to the world of geometricaltheorems but it is access that is mediated by features of the software en-vironment, certainly in the vital early and intermediate stages of using thesoftware. The research described in this paper illustrates that with care-fully designed tasks, sensitive teacher input, and a classroom environmentthat encourages conjecturing and a focus on mathematical explanation,students can make progress with formulating mathematical explanationsand coming to terms with inclusive definitions, both important aspectsof developing a facility with deductive reasoning. Without such factors,the mediational impact of the software could be such that it may distractstudents from the geometry of the problem situation or possibly reduce theperceived need for deductive proof.

De Villiers (1998) suggests that a focus on mathematical explanationis part of the route to a greater appreciation of the role and function ofmathematical proof. This study was designed to illuminate the impact ofdynamic geometry software on one small component of this, the relation-ships between quadrilaterals to form a hierarchical classification. Whilethe study was informed by research on the van Hiele model of thinking ingeometry, the results should not be taken as evidence, necessarily, of thevalidity of the van Hiele model. According to Duval (1998), for instance,

82 KEITH JONES

a model of mathematics learning in which different ways of mathematicalreasoning are organised according to a strict hierarchy (as in the van Hielemodel) is inappropriate. Rather than being representative of higher (orlower) levels of thinking, Duval argues that different kinds of cognitiveactivity have their own specific and independent development (this mayaccount for evidence that students can appear to be operating at more thanone van Hiele level simultaneously, a finding reported by Gutiérrez, Jaimeand Fortuny, 1991).

In the mathematics classroom, the practical issues of when and howto use dynamic geometry software are very important. Much previous re-search with dynamic geometry software has focused on students in uppersecondary school where the students have received considerable teachinginput in plane geometry, including the proving of elementary theorems,but are new to the particular software tool. The study reported in this paperfocuses on students in lower secondary school where students have quitelimited experience of the formal aspects of geometry (and have certainlynever seen a proof or been asked to prove a theorem). The evidence presen-ted in this paper confirms that students can make progress towards math-ematical explanations, which, it is suggested, should provide a foundationon which to build further notions of deductive reasoning in mathematics.

ACKNOWLEDGEMENTS

I would like to thank the school staff and students where the empiricalstudy was carried out, especially the teacher who helped with the designof the classroom tasks, and the anonymous reviewers of this paper fortheir insightful comments. The empirical work was part-supported by grantA94/16 from the University of Southampton Research Fund.

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Centre for Research in Mathematics Education,c/o Research and Graduate School of EducationUniversity of Southampton,Southampton, SO17 1BJ,UK

RAMÓN MARRADES and ÁNGEL GUTIÉRREZ

PROOFS PRODUCED BY SECONDARY SCHOOL STUDENTSLEARNING GEOMETRY IN A DYNAMIC COMPUTER

ENVIRONMENT

ABSTRACT. As a key objective, secondary school mathematics teachers seek to improvethe proof skills of students. In this paper we present an analytic framework to describeand analyze students’ answers to proof problems. We employ this framework to investigateways in which dynamic geometry software can be used to improve students’ understandingof the nature of mathematical proof and to improve their proof skills. We present the resultsof two case studies where secondary school students worked with Cabri-Géomètre to solvegeometry problems structured in a teaching unit. The teaching unit had the aims of: i)Teaching geometric concepts and properties, and ii) helping students to improve their con-ception of the nature of mathematical proof and to improve their proof skills. By applyingthe framework defined here, we analyze students’ answers to proof problems, observe thetypes of justifications produced, and verify the usefulness of learning in dynamic geometrycomputer environments to improve students’ proof skills.

KEY WORDS: Cabri, Dynamic geometry software, Computer learning environment, Geo-metry, Justification, Proof, Secondary school, Teaching experiment

1. INTRODUCTION

One of the most interesting and difficult research fields in mathematicseducation concerns how both to help students come to a proper understand-ing of mathematical proof and enhance their proof techniques. Over pastdecades, numerous researchers have experimented with different formsof teaching. Generally, we can say that their attempts to teach formalmathematical proof to secondary school students (frequently during shortperiods of time) were not successful (Clements and Battista, 1992). Thisobservation coheres with Senk’s research (1989) on the van Hiele model.She shows that most students who finish secondary school achieve onlythe first or second van Hiele level, and that progress from the second to thefourth level is very slow. Generally, it takes several years for students toreach level four from level two.

The work of Bell (1976a) and De Villiers (1990 and 1996) has ledto general agreement on the main objectives of mathematical proof: Toverify or justify the correctness of a statement, to illuminate or explain


88 RAMON MARRADES AND ANGEL GUTIERREZ

why a statement is true, to systematize results obtained in a deductive sys-tem (a system of axioms, definitions, accepted theorems, etc.), to discovernew theorems, to communicate or transmit mathematical knowledge, andto provide intellectual challenge to the author of a proof. However, stu-dents rarely identify with any of these objectives. We vitally need to knowstudents’ conception of mathematical proof in order to understand theirattempts to solve proof problems.1 That is, we need to know what it is forthem to ‘prove’ a statement or, in other words, what kind of argumentsconvince students that a statement is true. This knowledge can then be putto use in teaching a conception of mathematical proof that comes closer tothe conception currently accepted by mathematicians.

Along this line, the approach of mathematics education researchersto this topic has changed during recent years: The goal of educationalresearch is no longer attempting to find ways to promote skill in formalmathematical proof, but to study the evolution of the students’ understand-ing of mathematical proof and to find out how to help them improve theirunderstanding. This change of goals arises in part from the general convic-tion that secondary school students are not able to begin an apprenticeshipin methods of formal proof suddenly, as has sometimes been attempted(Senk, 1985; Serra, 1989). Instead apprenticeship in the methods of formalproof should be considered the last step along a long road.

Several authors have observed, from different points of view, studentsas they attempt to solve proof problems. Some authors have describedtypes of students’ justifications. Others have analyzed the ways in whichstudents produce justifications, including the ways in which they produceconjectures when required. A complete assessment of students’ justifica-tion skills has to take into consideration both products (i.e., justificationsproduced by students) and processes (i.e., the ways in which students pro-duce their justifications). In section 2 of this paper we describe the mainresults of previous research and integrate these results into a wider frame-work which considers both the ways in which students produce conjecturesand justifications, and the resulting justification itself.

Modern dynamic geometry software (DGS) has stimulated researchon students’ conceptions of proof by opening up new directions for thisresearch to take. The contribution of DGS is two-fold. First, it providesan environment in which students can experiment freely. They can easilycheck their intuitions and conjectures in the process of looking for pat-terns, general properties, etc. Second, DGS provides non-traditional waysfor students to learn and understand mathematical concepts and methods.These ways of learning pose many questions that mathematics educationresearchers should investigate.

PROOFS WITH DYNAMIC GEOMETRY SOFTWARE 89

In section 3 we describe an experiment in which secondary school geo-metry was taught using Cabri-Géomètre (Baulac, Bellemain and Laborde,1988). Cabri was used in the 30 activities of the teaching unit. In section4 we report on two case studies of two pairs of students. Our analysis ofthe solutions of both pairs and of their responses during clinical interviewsshow that each pair differed from the other in how the same proof problemswere solved. Finally, section 5 summarizes the main hypothesis of ourstudy and its conclusions, and raises some issues for future research.

Terms such as explanation, verification, justification, and proof havebeen used in the literature to refer, in one way or another, to convincing aspeaker, or oneself, of the truth of a mathematical statement. Sometimesthe same term carries more than one meaning (see, for example, the mean-ings of ‘justification’ in Bell, 1976a; Balacheff, 1988a; Hanna, 1995). Thisissue is beyond the scope of this paper. From now on in this paper, we willuse the term justification to refer to any reason given to convince people(e.g., teachers and other students) of the truth of a statement, and we willuse the term (formal mathematical) proof to refer to any justification whichsatisfies the requirements of abstraction, rigor, language, etc., demanded byprofessional mathematicians to accept a mathematical statement as validwithin an axiomatic system.

2. IDENTIFICATION OF AN ANALYTIC FRAMEWORK

There are many studies dealing with the processes by which students learnto justify mathematical statements. Some of these studies develop inter-esting, if partial, methods of analyzing the processes. These methods fitinto two main categories: Descriptions of forms of students’ work whensolving proof problems (Arzarello et al., 1998a; Balacheff, 1988a and b;Bell, 1976a and b; Harel and Sowder, 1996; Sowder and Harel, 1998), anddescriptions of students’ beliefs when deciding whether they are convincedby an argument about the truth of a statement, or not (De Villiers, 1991;Harel and Sowder, 1996; Sowder and Harel, 1998). Our study follows thefirst approach. In the second part of this section we describe an integratedframework which we later use to study students’ attempts to solve proofproblems. This framework provides a way to analyze and classify the pro-cesses of coming up with conjectures (when required by the problem)and of producing justifications, as well as analyzing and classifying thejustifications themselves.

Bell (1976a and b) identified two categories of students’ justificationsused in proof problems: Empirical justification, characterized by the use ofexamples as element of conviction, and deductive justification, character-


ized by the use of deduction to connect data with conclusions. Within eachcategory, Bell identified a variety of types: The types of empirical answerscorrespond to different degrees of completeness of checking the state-ment in the whole (finite) set of possible examples. The types of deductiveanswers correspond to different degrees of completeness of constructingdeductive arguments.

Balacheff (1988b) distinguished between two categories of justifica-tion, which he called pragmatic and conceptual justifications. Pragmaticjustifications are based on the use of examples, or on actions or showings,and conceptual justifications are based on abstract formulations of prop-erties and of relationships among properties. The category of pragmaticjustifications includes three types: Naive empiricism, in which a statementto be proved is checked in a few (somewhat randomly chosen) examples;crucial experiment, in which a statement is checked in a careful selectedexample; and generic example, in which the justification is based on opera-tions or transformations on an example which is selected as a characteristicrepresentative of a class. In this case, operations or transformations onthe example are intended to be made on the whole class. The categoryof conceptual justifications includes thought experiment, in which actionsare internalized and dissociated from the specific examples considered, andsymbolic calculations from the statement, in which there is no experimentand the justification is based on the use of and transformation of formalizedsymbolic expressions.

Harel and Sowder (1996), and Sowder and Harel (1998) identified threecategories of justifications (labelled proof schemes): Externally based, whenthe justification is based on the authority of a source external to students,like teacher, textbook, etc.; empirical, when the justification is based solelyon examples (inductive type) or, more specifically, drawings (perceptualtype), analytical or theoretical, when the justification is based on genericarguments or mental operations that result in, or may result in, formalmathematical proofs. Such arguments or operations can be based on gen-eral aspects of a problem (transformational type) or contain different re-lated situations, resulting in deductive chains based on elements of anaxiomatic system (structural or axiomatic type).

The above categories describe students’ outcomes (justifications) butthey do not consider the process of production of such outcomes. Fur-thermore, the focus of each study was different from that of the otherstudies, and each study was partial: With regard to the empirical/pragmaticcategories, Bell analyzed only the completeness of sets of examples usedby students; Balacheff focused on students’ reasons for selecting examplesand on how they used them; and Sowder and Harel differentiated justific-


ations based only on visual or tactile perception and on the observation ofmathematical properties.

Among the deductive/conceptual/analytical categories, those definedby Bell differ in the mathematical quality of their deductive chains. Sowderand Harel described two types of analytical justifications, one based ontransforming the conditions of the problem, and the other based on the useof elements of an axiomatic system. Balacheff identified only a type ofconceptual justifications (those which take into account specific examplesbut are not based on them as elements of conviction) and justificationsthrough symbolic calculations.

To promote progress in the description and understanding of students’answers to proof problems, we have defined a three-faceted classificationscheme in which all of the student’s activity – generation of a conjecture(if required), devising a justification, and the resulting justification – isconsidered:

1) Like Bell, Balacheff, and Sowder and Harel, we have differentiatedbetween two main categories, empirical and deductive justifications,depending on whether the justification consists of checking examples,or not.

2) Empirical justifications have been split into several subclasses de-pending on the ways students select examples to be used in theirjustifications, and each subclass has several types corresponding todistinct ways students use the selected examples in their justifications.

3) Deductive justifications have been split into two subclasses depend-ing on whether students select an example, or not, to help organizetheir justification, and each subclass has been divided into two typesdepending on the styles of deduction made to organize justifications.

The whole classification scheme is as follows:* Empirical justifications, characterized by the use of examples as themain (maybe the only) element of conviction: Students state conjecturesafter having observed regularities in one or more examples; they use theexamples, or relationships observed in them, to justify the truth of theirconjecture. When the conjecture is included in the statement of a prob-lem, students have only to construct examples to check the conjectureand justify it. Within empirical justifications, we distinguish three classes,depending on the way examples are selected:

– Naive empiricism, when the conjecture is justified by showing thatit is true in one or several examples, usually selected without a spe-cific criterion. The checking may involve visual or tactile perceptionof examples only (perceptual type) or may also involve the use of


mathematical elements or relationships found in examples (inductivetype).

– Crucial experiment, when the conjecture is justified by showing thatit is true in a specific, carefully selected, example. Students are awareof the need for generalization, so they choose the example as non-particular as possible (Balacheff, 1987), although it is not consideredas a representative of any other example. Students assume that theconjecture is always true if it is true in this example. We distinguishseveral types of justifications by crucial experiment, depending onhow the crucial example is used:Example-based, when the justification shows only the existence of anexample or the lack of counter-examples; constructive, in which thejustification focuses on the way of getting the example; analytical,in which the justification is based on properties empirically observedin the example or in auxiliary elements; and intellectual, when thejustification is based on empirical observation of the example, but thejustification mainly uses accepted properties or relationships amongelements of the example. Intellectual justifications show some decon-textualization (Balacheff, 1988b), since they include deductive partsin addition to arguments based on the example.

The main difference between analytical and intellectual justificationsis the source of properties or relationships referred to: In analyticaljustifications they are originated by the empirical observation of ex-amples (for instance, a student makes some measurements on an equi-lateral triangle and he/she notes that an angle bisector bisects theopposite side), while in intellectual justifications the empirical ob-servation induces the student to remember a property that had beenlearned before (for instance, the student makes the same measure-ments on an equilateral triangle and he/she remembers that its anglebisectors are also its medians).

The two main differences between a crucial experiment and naiveempiricism are i) the status of the specific example, and ii) that anexample used in a crucial experiment has been selected to be repres-entative of a certain class.

– Generic example, when the justification is based on a specific ex-ample, seen as a characteristic representative of its class, and thejustification includes making explicit abstract reasons for the truthof a conjecture by means of operations or transformations on the ex-ample. The justification refers to abstract properties and elements ofa family, but it is clearly based on the example. The four types of jus-tifications (example-based, constructive, analytical and intellectual)


defined for the crucial experiment are found here too, in descriptionsof how the generic example is used in the justification.The main difference between a crucial experiment and a generic ex-ample is that, in a crucial experiment, justification consists only ofexperimental verification of the conjecture in the selected examplewhile, in a generic example, justification includes references to ab-stract elements or properties of the class represented by the example.

– Failed answer, when students use empirical strategies to solve a proofproblem but they do not succeed in elaborating a correct conjecture orthey do state a correct conjecture but they do not succeed in providingany justification.

* Deductive justifications, characterized by the decontextualization of thearguments used, are based on generic aspects of the problem, mental oper-ations, and logical deductions, all of which aim to validate the conjecturein a general way. Examples, when used, are a help to organize arguments,but the particular characteristics of an example are not considered in thejustification. Within deductive justifications, we distinguish three classes:

– Thought experiment, when a specific example is used to help organ-ize the justification. Sometimes a thought experiment has a temporaldevelopment (Balacheff, 1988b), as a consequence of the observa-tion of the example, and it refers to actions, but these are internal-ized and detached from the example. Following Harel and Sowder(1996), we can find two types of thought experiments, depending onthe style of the justification: Transformative justifications are based onmental operations producing a transformation of the initial probleminto another equivalent one. The role of examples is to help foreseewhich transformations are convenient. Transformations may be basedon spatial mental images, symbolic manipulations or construction ofobjects. Structural justifications are sequences of logical deductionsderived from the data of the problem and axioms, definitions or ac-cepted theorems. The role of examples is to help organize the steps indeductions.

– Formal deduction, when the justification is based on mental opera-tions without the help of specific examples. In a formal deductiononly generic aspects of the discussed problem are mentioned. It is,therefore, the kind of formal mathematical proof found in the worldof mathematics researchers. We may also find the two types of justi-fications (transformative and structural) defined in the previous para-graph.

– Failed, when students use deductive strategies to solve proof prob-lems but they do not succeed in elaborating a correct conjecture or


Figure 1. Types of justification.

they elaborate a correct conjecture but they fail in providing a justi-fication.

Figure 1 summarizes previous types of justifications. This classificationis detailed enough to make a fine discrimination among a student’s answersto different problems. The two types of failed justifications are necessaryto complete the classification because the assessment of students’ justi-fication and proof skills cannot be associated only to correct solutionsof problems. Apart from classifying students’ answers, this classificationscheme is useful to evaluate the improvement of a student’s justificationskills in a learning period. The use of this classification to analyze datafrom our teaching experiment allowed us to evaluate changes in students’justification skills. Another application of this classification scheme couldbe to observe the possible influence of peculiarities of a specific envir-onment on students’ learning; for instance, it has been argued that DGSenvironments tend to promote some types of empirical justifications andinhibit formal justifications (Chazan, 1993; Healy, 2000).

The different classifications of justifications described in this section,including ours, implicitly assume that students work in a coherent linearway from beginning to end of the solution of a problem. However, thereality is, in many cases, different. Typically, many students begin by usingempirical checking and, when they have understood the problem and theway to justify the conjecture, they continue by writing a deductive justi-fication. It is also usual to make several jumps among deductive and em-


pirical methods during the solution of a problem. Arzarello et al. (1998a)considered these cases by analyzing the solution of problems paying spe-cial attention to the moment when the solver moves from an ascendingphase, characterized by an empirical activity aiming to better understandthe problem, generate a conjecture, or verify it, to a descending phase,where the solver tries to build a deductive justification. When solvingcomplex proof problems, often students move forth and back between bothphases. Therefore, these researchers’ proposal is to observe and analyzethe whole process of solution of proof problems, including early stepstoward identification of a conjecture or the finding of a justification. Anapplication of this construct to students working in a Cabri environmentcan be seen in Arzarello et al. (1998b). By merging the model proposedin Arzarello et al. (1998a) with the classification scheme defined above(Figure 1), we get a framework with two appraisal viewpoints to analyzesolutions to proof problems, where one of them corresponds to types ofjustification produced by students, and the other to shifts among empiricaland deductive methods taking place during the process of solution of prob-lems. In this way both the solution to a problem and the process of workingout such solution are analyzed together.

3. THE STUDY

The study reported here consisted in the design of a geometry teachingunit based on Cabri, its implementation in a mathematics class, and theobservation of students. In this paper we present the observation of twopairs of students. The main objective of the study was to investigate howDGS environments can help students improve their conception of proof inmathematics and their methods of justification.

DGS helps teachers create learning environments where students canexperiment, observe the permanence, or lack of permanence, of mathem-atical properties, and state or verify conjectures much more easily thanin other computational environments or in the more traditional setting ofpaper and pencil. The main advantage of DGS learning environments overother (computational or non-computational) environments is that studentscan construct complex figures and can easily perform in real time a verywide range of transformations on those figures, so students have access toa variety of examples that can hardly be matched by non-computational orstatic computational environments. A hypothesis of this study is that theCabri environment we have designed is more helpful than an environmentbased on non-computer didactical tools or on the traditional blackboard-


and-textbook, because the Cabri environment favours classroom organiza-tion to promote active methodologies.

The use of DGS to help students improve their ways of justificatingor proving in mathematics is controversial. Its supporters underline itsmultiple virtues as facilitator of learning and understanding (De Villiers,1998). On the other side, some researchers warn against the possibility thatthese environments may impede student’s leaving empirical justificationsto learn more formal methods of proof, because it is so easy to make use ofexhaustive checking on the screen that many students become convincedof the truth of conjectures and do not feel the necessity of more abstractjustifications (Chazan, 1993; Healy, 2000). In such cases, the teacher’srole is to help them go beyond, since research shows that an adequateplanning of activities in a DGS environment can help students produceabstract deductive justifications or, in particular, proofs (Mariotti et al.,1997; Mariotti, this issue). Another hypothesis of our study is that theCabri environment we have designed does not impede the improvementof students’ justification skills. On the contrary, this DGS environmentmay help students use different types of justification, setting the basis forthem to move from the use of basic to more complex types of empiricaljustifications, or even to deductive ones, as reflected by a change in thetypes of justifications produced in the experiment, and by a more coherentoscillation between ascending and descending phases.

In most research on teaching in DGS environments, participant studentswere novice users of the software, so part of the time in those experimentswas devoted to teaching them how to use the software. Furthermore, stu-dents’ lack of experience in the use of software caused many of them touse wrong strategies to solve problems, or strategies more naive than whatwould have been used in a more familiar environment. We have eliminatedthis possible limitation from our study, because participant students hadused Cabri over several months in the previous academic year, so they wereknowledgeable of the software, and they understood the meaning of theactions to be accomplished with Cabri (dragging, modification of objects,etc.). They also understood the difference between a figure as an objectcharacterized by mathematical properties implicit in commands used forits construction, and a drawing as a particular representation of a figure onthe screen2 (Parzysz, 1988; Laborde and Capponi, 1994).

3.1. Sample

A group of 16 students in their 4th grade of Secondary School (aged 15–16 years) participated in the teaching experiment. It was carried out aspart of the ordinary mathematics teaching, with their own teacher (one


of the researchers and authors) and during the standard class time. Theclassroom had a set of PC computers with Cabri-Géomètre (version 1.7).Students worked in pairs. This group of students had the same math teacherthe year before, when they began to work with Cabri to solve conjectureproblems, so the teaching experiment could be organized on the basis oftheir experience and knowledge gained during the previous year.

Two pairs of students were selected by the teacher before beginning theexperiment for follow-up in this case study. These four students, all boys,represented abilities and attitudes from high to average. One of them wasthe best in the class and the other three were average (it was decided notto include students whose reasoning skills were judged to be very poor, someaningful data collection would most probably not be possible).

3.2. The teaching experiment

This teaching unit was part of the normal content of the course, and stu-dents need to pass an exam at the end of the course. The teaching unit hadas main objectives:

– To facilitate the teaching of concepts, properties and methods usuallyfound in the school plane geometry curriculum: Straight lines andangles among them. Properties and elements of triangles (perpen-dicular bisectors, angle bisectors, etc.). Congruence and similarityof triangles. Relationships among angles and/or other elements of atriangle. Quadrilaterals, their properties and elements. Classificationsof triangles and quadrilaterals. Circles, angles and tangents.

– To facilitate a better understanding by students of the need for andfunction of justifications in mathematics.

– To facilitate and induce the progress of students toward types of jus-tification closer to formal mathematical proofs. In terms of van Hielelevels, with respect to justifications, the objective was to help studentsto do, by the end of the experiment, justifications in, at least, the thirdlevel.

The teaching unit had 30 activities. Each activity was structured in severalphases, beginning with a phase where students had to create a figure inCabri and explore it (in a few activities the figure was provided by theteacher in a file to be opened by the students). In the second phase studentshad to generate conjectures (in some activities, the students were askedonly to check a given conjecture). In the last phase students had to justifyconjectures they had stated (some activities did not include this phase).The aim of activities 1 to 11, 14, and 22 was to teach several geometryconcepts necessary, as previous knowledge, to solve activities 12 to 30.


Those activities did not include the phase of justification of conjectures.Annex 1 includes summarized information about the activities. Each activ-ity was presented to the students in a worksheet where they had to writetheir observations, comments, conjectures and justifications.

The activities were conceived in an endeavor to get maximum benefitfrom the dynamic capability of Cabri. As usual in most Cabri environ-ments, in this teaching unit dragging had a central role in the generationand checking of conjectures: As part of the didactical contract present inthe teaching experiment, the ‘dragging test’ acquired the status of an essen-tial element to check the validity of a construction, since students verifiedthat a figure was correct because it passed the dragging test, i.e. they couldnot mess the figure up by dragging (Noss et al., 1994). Furthermore, drag-ging was a very helpful tool for students when they had to check or stateconjectures (they could easily recognize regularities that they identifiedas mathematical properties) and to make empirical justifications. Manyactivities would have been too difficult for these students if stated in apaper-and-pencil environment, because they could only be solved by usingdeductive reasoning far from most students’ capability (e.g., activity 20;see section 4.2). Other activities could not have been solved with paperand pencil by any student (e.g., construction 1 in activity 30; see section4.3) because they lacked the necessary knowledge of geometrical facts andrelationships, and abstract reasoning ability.

Dragging was sufficient to convince most pupils of the correctness ofconjectures, so questions like ‘why is the construction valid?’ or ‘why isthe conjecture true?’ were important to induce students to elaborate jus-tifications beyond the simple checking of some examples on the screenby dragging. As part of the didactical contract defined in the class, pupilsknew that requirements like ‘justify your conjecture’ carried the implicitmeaning of ‘justify why your conjecture or construction is true’.

Two 55-minute mathematics classes per week were devoted to the teach-ing experiment. Students worked on each activity during two consecutiveclasses, so the experiment lasted about 30 weeks. During the first classof an activity, the pairs of students worked autonomously in solving theactivity. The teacher observed their work and answered their questions. Bythe end of this class, each pair had to give the teacher their results writtenon the worksheets, and also had to save their constructions in computerfiles. Each pair had to write one answer, agreed by both students. At thebeginning of the second class, the teacher gave students a list with theirdifferent answers to the problem, and several students (selected by theteacher) presented their solutions to the group. Then, the class, guidedby the teacher, discussed the solutions presented, the correctness of the


conjectures and the validity of their justifications. Finally, the teacher madea summary of the activity and stated the new results students had to learn.

In mathematics, students usually need help to recall all results learntin preceding classes that may be used, or have to be used, to elaboratedeductive justifications in subsequent problems. Often they cannot solve aproblem because they do not remember a key result. To reduce this prob-lem in the teaching experiment, each student had a ‘notebook of acceptedresults’ consisting in lists of previously learnt axioms, definitions, proper-ties and theorems. In this way, students could consult their notebook whenthey did not recall a result. After each activity, the new accepted resultslearnt in the activity were added to notebooks.

3.3. Methodology of data gathering

Three ‘test activities’ (activities 12, 20 and 30) were selected from theteaching unit to be a source of detailed information about students’ ways ofconjecturing and justifying. These activities were selected because: Activ-ity 12 was the first one where students were asked to justify their con-jectures. Activity 20 was a proof problem situated after two thirds of theteaching unit. Activity 30, also a proof problem, was the last activity inthe teaching unit. The information gathered to analyze students’ activityduring this teaching experiment came from several sources:

– The answers to the test activities written by the two pairs of studentson their worksheets, plus the files with constructions made in Cabri.The command ‘History’ lets us see how a figure has been constructedand, in some cases, it helps us identify previous attempts discardedby students.

– To record interactions with Cabri of the two pairs of students, thecommand ‘Session’ was used (Cabri saves in the hard disk a snapshoteach time the screen is modified, and the sequence of snapshots canbe viewed like an animation).

– Three semi-structured clinical interviews (Malone, Atweh and North-field, 1998) to the two pairs of students selected. After each test activ-ity, the teacher (also researcher) interviewed each pair, asking themquestions related to their answers to the test activity. During inter-views, students had access, if necessary, to the notebook of acceptedresults, their worksheets and their computer files. They also coulduse Cabri to explain their answers, to try again to solve the activ-ity, etc. The clinical interviews were video-recorded, and afterwardtranscribed for subsequent analysis.


4. DATA ANALYSIS AND RESULTS

The reduced number of students participating in the experiment, the waythe research had been organized, and the kind of data collected suggesta qualitative case study analysis of the experiment is most reasonable. Inthis section we present the cases of the two pairs of students mentioned insection 3.3. We cannot analyze here these students’ answers to all the activ-ities in the teaching unit, due to space limitation. We centre the analysis inthe three test activities and subsequent clinical interviews, since these areenough to observe any change in students’ justifications throughout theteaching unit, in relation to the third objective stated in section 3.2.

In the following paragraphs we summarize the protocols of students’solution of the test activities, based on records of the command Session,answers on worksheets, and Cabri files saved in the computer. This inform-ation is clarified with answers given during clinical interviews. Afterwardwe compare, for each pair of students, the information from each test activ-ity, and get conclusions about their conception of proof. Text inside square[brackets] in protocols was added to clarify the meaning of students’ an-swers. In particular, we labelled points used by students but not labelledby them. Round (brackets) in protocols were written by the students.

4.1. First test activity

The statement of the first test activity (activity 12) was:A, B, and C are three fixed points. What conditions have to be satisfied

by point D for the perpendicular bisectors to the sides of ABCD to meet ina single point? (Figure 2)

Figure 2.

4.1.1. First case (students H and C)(1) H and C first built a convex quadrilateral with the perpendicular bi-

sectors of its sides, and dragged it. They made many transformations


Figure 3.

Figure 4.

to the quadrilateral without any apparent positive result. Then they ad-ded the measures of angles and sides of the quadrilateral and draggedit again. They obtained only one quadrilateral with a single meetingpoint, a rectangle.

(2) H and C continued dragging, and they got a crossed-sides quadrilat-eral whose perpendicular bisectors almost met in a single point [Fig-ure 3]. After this example, they continued dragging and got severalcrossed-sides quadrilaterals verifying the condition of the problem.

(3) H and C worked again with convex quadrilaterals. They got a quadri-lateral [Figure 4] and several rectangles with a single meeting point,and other quadrilaterals where the perpendicular bisectors almost metin a single point.

(4) H and C transformed the quadrilateral into a triangle by superimpos-ing two consecutive vertices, B and C [Figure 5]. As students werenot accurate, B and C did not coincide exactly, so Cabri continuedshowing four perpendicular bisectors that met in a single point. Bydragging A or D, they transformed the ‘triangle’, but again the fourperpendicular bisectors did not meet in a single point.


Figure 5.

Figure 6.

(5) H and C desisted in the exploration of ‘triangles’. After more drag-ging, students got several convex non-rectangular quadrilaterals toverify the condition. Then, they stated a conjecture: “The sum ofangles A and C is equal to the sum of [angles] B and D if we wantperpendicular bisectors to meet [in a single point]. The sum of theangles [in each pair, A+C and B+D] is 180◦.”

(6) H and C constructed a circle with centre in the intersection pointof two perpendicular bisectors through vertex C. Vertex D was alsoon the circle, but vertices A and B were not [Figure 6a]. Then theymoved vertices A and B onto the circle [Figure 6b]. H and C wrote ajustification: “The perpendicular bisectors meet in a point. That pointis the centre of the circumscribed circle. The vertices are equidistantfrom the centre of the circle.”

The figure (a quadrilateral) which H and C made was a generic examplethat they transformed, by dragging, into many different drawings ((1) to(3) and (5)). In the interview, students explained how their conjectureemerged: “We made many [convex] quadrilaterals and we added them


[opposite angles] every time. We noted that they had some relationship.”This was the ascending phase of the solution.

H and C were not able to use that relationship in their justificationbecause they still had not learnt properties of angles in a circle and re-lationships among them (such properties were learnt in activity 29). WhenH and C wrote (6), they did not refer to the conjecture they had stated but,implicitly, they produced another conjecture, namely, that if the verticesof the quadrilateral are on the circumscribed circle, then all perpendicularbisectors meet in the centre. There was not a logical relationship betweenH and C’s conjecture (5) and their justification (6), so they were forced toformulate a justification based on other properties. When, in the interview,they were asked to justify why perpendicular bisectors meet in the centreof the circumscribed circle, they answered: “We make the circle”, and theyrepeated the construction they had made in the classroom (6). Most likely,H and C drew the circle because they had associated this problem to thecase of perpendicular bisectors of a triangle (activity 8), as a consequenceof their work with ‘triangles’ in (4).

H and C wrote in (6) a justification that shows their switch to the des-cending phase, although this is not clearly related to the previous ascendingphase. It is an empirical justification, since it came from the handling andobservation of examples, it was based on observed facts, and it mentionedproperties observed in examples. Students tried to express a conjecturedecontextualized from the examples observed, but they did not make anyabstract deduction, because they always referred to drawings on the screento try to justify their conjecture. Thus, this is an example of empiricaljustification by analytical generic example.

4.1.2. Second case (students T and P)

(1) T and P first created a convex quadrilateral, without perpendicular bi-sectors of its sides, and they dragged it for a while passing a draggingtest. Then they dragged the quadrilateral until they got a rectangleand, after measuring the sides, a square. Then they constructed theperpendicular bisectors of the sides. By dragging, they got severalquadrilaterals with perpendicular bisectors meeting in a single point.T and P wrote on their worksheet: “The perpendicular bisectors meetin a single point in squares and also in some other quadrilaterals, butnot in all.”

(2) T and P marked intersection points of two pairs of perpendicular bi-sectors and added the measure of the angles. Then they looked formore shapes verifying the condition, by making very short draggingsthat produced ‘quasi-square’ quadrilaterals with all angles measur-


Figure 7.

ing between 88◦ and 92◦ and quasi-congruent sides. Now T and Pmade longer draggings, so they produced a set of very different draw-ings, including a crossed-sides quadrilateral and several rectangles.The only cases with perpendicular bisectors meeting in a single pointwhere rectangles. Then students raised a conjecture: “In principle,the condition [for perpendicular bisectors to meet in a single point] isthat with D the quadrilateral has all right angles (90◦).”

(3) T and P continued dragging to check their conjecture, until they founda counter-example [Figure 7]. This forced them to complete their con-jecture: “But it [the property of meeting in a single point] is also truewhen there are two acute angles and two obtuse angles. Furthermore,acute angles are consecutive, and also obtuse angles.”

(4) T and P continued dragging to check their new conjecture, and theyfound some counter-examples [Figure 8], so they modified their con-jecture: “We have found a new conclusion [conjecture]: The differ-ence among obtuse [angles] and [among] acute [angles] has to be thesame.” The students dragged the figure a bit more and they consideredtheir work at an end.

The conjecture stated in (1) was derived from examples obtained by drag-ging. When counter-examples appeared, the conjecture was refined in (2).Conjectures in (1) and (2) referred mainly to squares and rectangles, re-spectively. Although T and P had found other quadrilaterals with perpen-dicular bisectors verifying the condition (as seen in the first conjecture),they were looking for a standard family of quadrilaterals as a solution.For this reason, when they found the counter-example in Figure 7, theycould not improve their conjecture again, and they were forced to look fora completely different one (3). Again, after new counter-examples werefound (Figure 8), students improved their conjecture in (3) by modifyingthe condition on the relationship among angles (4). Therefore, the process


Figure 8.

of getting conjectures was grounded on the observation of drawings andregularity in the measures of angles.

This protocol shows a clear example of activity in the ascending phaseand shows that students did not culminate by passing after (4) to the des-cending phase of elaboration of an abstract justification. This is not sur-prising, given that T and P had never been asked before to justify theirstatements in a deductive way. The Session record for this problem showedthat most of their dragging actions were not long aleatory movements, butvery short translations of vertices. This indicates that after stating eachconjecture, T and P used deliberately sought examples to check each con-jecture. In the interview the students stated: “Instead of moving the sides,we moved [the vertices] to make the two points [marked in (2)] cut [coin-cide]. And thus it was always the same, but moved a little and did not cut[did not coincide].”

In (1) to (3) students found counter-examples, but in (4) they did not, soafter the final dragging (end of (4)) they considered that their last conjec-ture was proved. Students explained in the interview after the teacher askedthem about the truth of the conjecture (4): “We did not find any counter-example.” This was the first problem in the teaching unit where studentshad to justify for themselves the truth of a conjecture they had elaborated.Hence, it should not be surprising that their attempts were not coordinated,were sometimes contradictory, and were not carried to a valid result, andthat they did not feel the necessity to articulate an abstract justification.Therefore, students implicitly justified the conjecture (4), and this justific-ation corresponds to the model of empirical justification by example-basedcrucial experiment.


4.2. Second test activity

The statement of the second test activity (activity 20) was:

Construct a shape (Figure 9) fitting the following conditions:1. Segment AB is parallel to segment CD (i. e., AB // CD).2. Segment AB has the same length as segment AC (i. e., AB = AC).

Figure 9.

Construct segment CB (Figure 10).

Figure 10.

Investigate: Is segment CB the angle bisector of � ACD?Justify your affirmative or negative answer to previous question. We

assume that your conclusion is true, but why is it true? It is necessary to usegeometric properties studied and accepted in the classroom.

4.2.1. First case (H and C)(1) H and C first created the figure requested. By dragging, they saw

that there was a mistake in their figure, and they corrected it. Thenew figure passed the dragging test. Then they measured � ACB and� BCD, and segments AB and AC. Then they used the dragging test,by moving C, to validate the stated conjecture.

(2) In an attempt to elaborate a justification, H and C added some aux-iliary elements: They constructed segment BD, measured segment


Figure 11.

Figure 12.

CD, and moved D so that ABDC had four equal sides. They recog-nized that this was a particular case of the figure they were asked toconstruct.

(3) H and C erased point D and segment BD, constructed the line paral-lel to AC through B, marked the point [K] of intersection with theline that goes through C, and constructed segment AK. They alsoconstructed the perpendicular bisector of AK, that coincided withsegment BC, so they hid it. H and C noted the division of ABKC intotwo congruent isosceles triangles. Finally, they hid line BK [Figure11].

(4) H and C constructed the line perpendicular to AB through K andmarked the point [V] of intersection with BC, and the point [M] ofintersection of AK and BC. They measured segments AK, KV, andAM [Figure 12]. Students dragged the figure and observed the valuesof measurements. They hid line VK, and measured segment MK.

(5) H and C noted that they could not drag K, since it was an intersectionpoint. Then, they erased segment AK, marked a point [D] on line CKand constructed segment AD [Figure 13]. By dragging, H and C notedthat the triangles contained in �ABC were different and they movedD so that those triangles looked congruent, i.e., when D coincidedwith K.

(6) H and C constructed the line AK as perpendicular bisector of BC, andmarked again the intersection point [M] of BC and AK. They set apartD and K, and measured several segments [Figure 14]. These meas-urements showed the congruence of �ACM, �ABM, and �CMK.


Figure 13.

Figure 14.

(7) H and C wrote on their worksheet the first part of their justificationfor the stated conjecture: “[�ABC] is isosceles. � ACB = � ABC. Weobtain two triangles [�AMC and �ABM]. Have a common side (CB)[they mean a congruent side: CM = BM]. The segments obtained byintersection in the parallel lines are equal [AB = AC].”

(8) H and C completed their previous justification: “We have an isosceles[triangle] (�ABC), we construct the perpendicular bisector that splitsit into two equal triangles [�AMC and �ABM]. The two oppos-ite triangles [�ABM and �CMK3] are equal, therefore: �AMC =�ABM = �CMK, so � KCM = � MCA. [�ABM and �CMK are con-gruent] because they have an equal angle (alternate interior) [� KCM =� MBA], [other] equal angle (opposite) [� CMK = � BMA] and a com-mon side [they mean a congruent side: CM = BM].”

H and C began to check the conjecture by using a dragging test (1), fol-lowed by a first attempt to find elements to elaborate a justification (2).That attempt was abandoned when they noted that rhombus ABDC was aparticular case of the figure. They did not note that such particularizationwas irrelevant for the justification of the conjecture, since they wanted toelaborate a justification valid for any point D. Afterward they tried againto elaborate a justification, by adding several auxiliary elements, makingmeasurements, and dragging to discover relationships ((3) to (6)).

The final part of students’ experimental work (6) helped them write ajustification ((7) and (8)), as indicated by their decision to elaborate thejustification on the basis of several congruent triangles they had identified


after watching measurements in Figure 14. To complete the justification(8), students used �ABM as an auxiliary object to make explicit the con-gruence of �AMC and ACMK. They took into consideration propertiesobserved during the dragging in (4) to (6). Students referred to these prop-erties in their attempts to form a deductive sequence, but their attemptslacked decontextualization (Balacheff, 1988a), since their justification wasmore a narrative of the construction ((5) and (6)) than a deduction fromhypothesis and accepted theorems or definitions. Therefore, this is an em-pirical justification by an analytical generic example.

The summary of the protocol shows clearly that students went fromthe ascending phase ((1) to (6)) to the descending one when they began toverbalize the justification (7). The need to write a justification was inducedby the didactical contract in the class that established the need of elaborat-ing justifications based on geometric properties previously accepted in theclass. In the clinical interview after this activity, pupils said they knew that,after completing the construction, “we had to pay attention to the acceptedrules.”

4.2.2. Second case (T and P)(1) T and P began the solution of this problem in the same way as H

and C. They also made some mistakes that were discovered duringa dragging test. After creating the correct figure, they measured AB,AC, � BCA and � BCD, and constructed segment BD to check if theconjecture was true in parallelograms. By dragging, T and P saw thatsometimes polygon ABDC was not a parallelogram, so they erasedBD and decided to abandon this focus.

During the clinical interview, T and P explained that they constructed BDbecause “the rule of the parallelogram, that these two triangles [�ABCand �BCD] are always equal.”

(2) T and P measured � ABC [Figure 15]. By dragging, they saw that� ABC was always congruent to � ACB and � BCD.

(3) A bit later, students justified the congruence of � ABC and � BCD:“ � BCD = � ABC because they are alternate interior angles. AB = AC.AB is parallel to CD.” This certainty, based on an accepted property,induced T and P to erase the measure of � ABC.

T and P believed that they could write a justification: “After having this[result], we try to prove that � ACB is equal to � ABC and we do it byconstruction.”

(4) T and P constructed the line perpendicular to CB through A, markedthe point M of intersection of this line with BC, and measured � CAM,


Figure 15.

Figure 16.

� BAM and � AMB [Figure 16]. Next, they checked if line AM wasthe angle bisector of � CAB by comparing, while dragging, � CAMand � BAM.

(5) Finally, T and P wrote on the worksheet their justification, as a con-tinuation of (3): “If AB = AC and AB is parallel to CD, then � BCD =� ABC (alternate interior) and � ACB = � ABC [because �ACM =�ABM] for the SAS criterion (AB = AC, AM is a common side,� CAM = � BAM). Therefore, if �ACM = �ABM then � ACB = � ABC.� ABC = � BCD and � ACB = � ABC → � BCD = � ACB ([so CB is the]angle bisector of � ACD).”

In this protocol we can differentiate two parts: First, T and P added someauxiliary elements to the figure and made several measurements ((1) to(4)). Eventually they found several pieces of information ((2) to (4)) thatthey organized in a proof (5). Their work in (2) to (4) was typical of theascending phase, where the problem is better understood and informationis gathered empirically. Students recall known theorems after seeing thebehavior of the drawings on the screen. T and P’s work in (5) is typicalof the descending phase, in which an attempt is made to put the collectedinformation into a deductive justification. So there was a full coherencebetween ascending and descending phases. This justification was clearlyorganized in a deductive argument, with almost all the statements justifiedby recall of pertinent accepted theorems. The only exception is that con-gruence of � CAM and � BAM was empirically verified in (4), but studentsnever justified it theoretically, since they did not note that �ABC was an


isosceles triangle with AM an altitude, and they used � CAM = � BAM toprove that �ACM was congruent to �ABM instead of using � AMC =90◦ = � AMB. Anyway, (5) is an empirical justification by intellectualgeneric example, since it is mainly based on accepted properties learnedpreviously.

In the previous solution, (1) to (4) are, as a whole, an ascending phase,although it is possible to identify several movements between ascendingand descending phases: In (1) there was an ascending phase that did notcrystallize in a descending one, since students abandoned the argument.In (2) there was a new ascending phase that shifted in (3) to a short des-cending phase when T and P explicitly recognised the property of alternateinterior angles and they decided that they could erase an auxiliary element.In (4) T and P moved back to the ascending phase, again jumping to thedescending phase when they began to write the justification (5).

4.3. Third test activity

The statement of the third test activity (activity 30) began by recalling theconcepts of tangent and secant lines to a circle. Then students were askedto make two constructions:

Construction 1: Construct a circle with centre O through point A. Mark apoint B in the circle. Construct secant line AB. Construct line OB and nameD the other point of intersection of OB and the circle. Measure � DBA.

Investigate and conjecture: Look at � DBA while you move point B alongthe circle. Which value does � DBA approach when point B is very near topoint A?

When point B is moved onto point A, line AB touches the circle in onlyone point, so AB is tangent to the circle. What is the relationship betweena line tangent to a circle and the radius to the tangency point? Justify yourconjecture.

Construction 2: Construct a circle with centre O. Mark a point P exteriorto the circle. Construct the tangent lines to the circle going through pointP. Describe the construction you have made.

Justify the correctness of your construction: Why is it correct? It is ne-cessary to use geometric properties studied and accepted in the classroom.

It is difficult to solve this problem if the way of connecting points O andP is not discovered (a circle with centre in the midpoint of OP; see Figure24). This technique was unknown to the students, but they had studied, inactivity 29, that any angle inscribed in a semicircle is a right angle, andthis property was included in the list of accepted results.

The definition of tangent to a circle known to the students was thatof a straight line touching the circle in only one point. The objective of


Figure 17.

construction 1 was to help the students discover the constructive charac-terization of a tangent to a circle as the line perpendicular to the radiusof the tangency point (Figure 17). The two pairs of students discoveredthis result easily; it was included as a theorem in the notebook of acceptedresults; and, as expected, they used it in the second part of the activity.Therefore we will focus the analysis in this paper only on construction 2.

4.3.1. First case (H and C)During the solution of this problem, H and C made a series of attempts toconstruct the required figure. All them were unsuccessful and ended eitherwhen their figure was messed up in a dragging test or when students gota drawing which fitted the main requirement of the problem (two tangentlines through P) but they abandoned it because they were aware that suchdrawing did not solve the problem. This series of attempts is interestingbecause each one is more perfect than previous ones and many of themgive students a new clue to the solution:

(1) H and C began by creating a circle with centre O and a point P exteriorto it. Then they constructed a line through P and another point exteriorto the circle. They moved P onto the circle and rotated the line to looktangent to the circle at P. They erased the figure.

(2) H and C constructed line OP and the line perpendicular to OP throughits intersection point with the circle located between O and P. Studentserased the perpendicular line, constructed a point [X] on the circle andline PX, and they moved X so that PX looked tangent to the circle.They erased the figure.

(3) H and C constructed a line through O and a point of the circle [Y],and the line perpendicular to OY through Y. This line, tangent to thecircle, passed very near to P, but a dragging test showed that P did


Figure 18.

Figure 19.

not belong to it. Then, they erased P and created it as a point on thetangent line instead of as a free point [Figure 18].H and C tried twice to construct the second tangent to the circlethrough P [Figure 19]. They considered the second drawing valid,although they knew that it did not solve the problem since P was nota free point.Now, H and C made three more attempts to construct the tangents,but neither of them passed the dragging test. Finally, they erased thefigure.

(4) H and C constructed again a circle and a free point P exterior toit, two lines through O, and two lines through P and the points ofintersection of the previous lines and the circle [Figure 20a]. Afterdragging, students erased the lines.H and C constructed line OP, circle with centre P and point O, and twolines through P and the points of intersection of the circles [Figure20b]. After dragging, students abandoned this figure, although theydid not erase it.H and C marked two points on the circle [almost symmetrical respectto OP], constructed their radii, constructed two lines through thesepoints and P, and marked the angle of a line and its radius. Then


Figure 20.

Figure 21.

students moved the points so that the lines looked tangent and theangle measured 90◦ [Figure 20c]. H and C erased this figure becausethey knew that it was not a solution.

(5) H and C constructed two free points P and [A], the circle with centreA and point P, line PA, a point [B] on the circle, the line perpendicularto PA through B, and the other point [C] of intersection of this lineand the circle [Figure 21a]. H and C also constructed several circlesthat were considered useless and erased. Then they constructed linesPB and PC, point O of intersection of PA and the circle, and linesperpendicular to PB through B and to PC through C. Students notedthat the two last lines met in O. Finally, they constructed the circlewith centre O and point B, and measured the right angles � PBO and� PCO [Figure 21b].

The first actions of H and C ((1) and (2)) were quite far from conditionsof the problem. Probably, the students had not understood the statement ofthe problem, and successive constructions in (1) to (3) corresponded to newreadings of the problem. This kind of initial or intermediate constructionduring the solution is frequent in difficult or complex problems like thisone. Usually figures are wrong or incomplete and they do not lead to ajustification, because they do not pass the dragging test. In the protocol ofthe second pair of students (T and P) below, this situation is also apparent.


This kind of activity, when successful, precedes empirical justifications byperceptual naive empiricism.

H and C’s actions in (3) indicate that they decided to try to solve avariation of the problem, by constructing a free tangent, P on it, and thenthe second tangent. H and C did not erase last figure in (3) (Figure 19b),and they used it as a reference from time to time while they continuedtrying to solve the problem. Sometimes, students stopped working on thenew figure and again manipulated that one. In (4) students tried severalconstructions, until they got one that they considered correct (5). H andC had ever in mind the property of a tangent line to be perpendicular toits radius, as they explained during the interview: “We start from this.That [tangents] had necessarily to be 90◦ [with the radius of the tangencypoint].” Their main difficulty was to find the tangency points. In their lastfigure (5) H and C constructed first the circle with centre A, then tangencypoints, and finally the circle with centre O.

The whole H and C’s activity corresponded to the ascending phasesince they only worked on understanding the problem and on trying toget some idea to help them to solve it. Figure 21b should have inducedthem to construct a correct figure and, therefore, to shift to the descendingphase. In the interview students said that they did not have time to write ajustification for the validity of their last figure, so the teacher asked them tojustify it verbally. H and C explained the process of construction and gavereasons for the successive steps in it, but they were not able to organizea coherent complete deductive justification, even though they knew theproperty of perpendicularity of a tangent line to the radius of the tangencypoint, and how to find the centre of the circle circumscribed to a righttriangle (midpoint of the hypotenuse). Therefore, H and C’s justificationfor their construction (5) was empirical by constructive generic example,since they tried to construct a generic figure (in (3) students rejected afigure because it was a specific example where P was not a free point) andbased their verbal justification mainly on the process of construction of thefigure.

4.3.2. Second case (T and P)

(1) T and P began construction 2 by creating a circle with centre O anda point P exterior to it. Then they constructed a line through P andother point exterior to the circle. Then, they moved P onto the circle,so that the line looked tangent, and they linked P to the circle with‘redefine an object’. By dragging, they noted that the line was notalways tangent to the circle. They erased the figure.


Figure 22.

(2) T and P constructed another circle with centre O, a point P exteriorto it, a point [B] on the circle, line OB and the perpendicular to OBthrough P [Figure 22a]. Then, they moved P so that the perpendicu-lar line passed through B and, therefore, it was tangent to the circle[Figure 22b]. Obviously, this figure did not pass the dragging test, sostudents erased it.

(3) T and P constructed two points O and [A], line OA, the line perpen-dicular to OA through A, and circle with centre O and point A. Thenstudents constructed a point [B] on the circle, line OB, and the lineperpendicular to OB through B. Finally, T and P marked the point [P]intersection of the two perpendicular lines [Figure 23], and draggedthe figure to observe it. Students noted that P was not a free point(it could not be dragged), but they continued observing this figurebecause PA and PB were always tangent lines.

(4) T and P started a new attempt by constructing a circle with centreO and a point P exterior to it. Next, they constructed segment OP,middle point X of this segment, circle with centre X and point O,points A and B, intersection of the two circles, and lines PA and PB[Figure 24]. Now they hid auxiliary elements and made a draggingtest. As the figure passed the dragging test, T and P thought that theyhad found the solution of the problem. They constructed radii OAand OB, marked � OAP and � OBP, and began the elaboration of ajustification.

(5) T and P wrote this justification on their worksheet: “We have usedthe property of the triangle inscribed in a [semi]circle [they meanthat any angle inscribed in a semicircle is a right angle]. From thedrawing [on the screen] we know that triangles AOP and BOP are[right triangles]. As we have proved before [construction 1], tangentsare perpendicular [to their radius] (90◦). Back to the beginning, wehad to look for right triangles to construct tangents, and we have used


Figure 23.

Figure 24.

the above mentioned property. By constructing a circle with diameterOP.”

We can see how T and P used different kinds of inductive or deductivereasoning, with increasing sophistication, while solving this problem. Wecan classify them according to the type of justification they would haveproduced: Since some points and lines were situated visually in the correctplace, in (1) and (2) we see an ascending phase typically associated withempirical justifications by perceptual naive empiricism. In (3) studentstook a step forward, since they created the figure based on a necessaryproperty of tangents. They constructed a figure very similar to the solutionasked (the difference is that P was not a free point). The figure let themobserve dynamic relationships among circle, straight lines, and points, andidentify invariants. In particular, they recognized the right triangles thatwere the key to make the correct construction (4). In fact, the constructionmade in (4) was a direct consequence of the analysis they made in (3).Since in (3) T and P looked for a particular drawing, they would be inan ascending phase associated to an empirical justification by constructivecrucial experiment.


TABLE I

Summary of students’ solutions to test activities

Activ. Students H and C Students T and P

Justification Phases Justification Phases

1st Empirical by analytical ↑↓ Empirical by example-based ↑generic example. crucial experiment.

2nd Empirical by analytical ↑↓ Empirical by intellectual ↑↑↓↑↓generic example. generic example.

3rd Empirical by constructive ↑ Empirical by intellectual ↑↓generic example. generic example.

↑ = ascending phase. ↓ = descending phase.

T and P were working in the ascending phase while they looked forthe way to construct the tangents ((1) to (3)). In (4) there was a shiftin their work, since they did not look for examples nor explore specificconfigurations any more, but constructed a figure that was a generic ex-ample of the construction required. So after (3) students had moved tothe descending phase. T and P completed the right construction (4) and acorrect justification (5). Therefore, this was an empirical justification byintellectual generic example. It was empirical because it came from theobservation and manipulation of some examples, and it was intellectualbecause students tried to decontextualize the justification, which was notdirectly based on the example, but on a known theorem.

4.4. Summary

Table I summarizes the analysis we have made of answers of the twopairs of students to the three test activities. We observe that, throughoutthe teaching unit, H and C continued to propose empirical justificationsby analytical or constructive generic examples. On the other hand, T andP, although they always elaborated empirical justifications, evolved pos-itively from an example-based crucial experiment to intellectual genericexamples.

Students’ movements from one phase of the solution of a problem toanother describe the process of solution, since such movements are relatedto their success in finding a correct answer. T and P’s solutions of the threeproblems are a clear example: In the first test activity, T and P were not ableto leave the ascending phase, since their work was based only on identi-fication of specific examples, and they did not find a valid conjecture. Inthe second test activity, they jumped several times between ascending and


descending phases, since they first justified an auxiliary property and laterthey justified their conjecture. In the third test activity, T and P only jumpedto the descending phase once, when they completed their experiments withspecific examples and began to construct the correct figure. In second andthird activities, T and P constructed several figures during the solution, butthe difference was that in the second test activity intermediate drawingshelped them discover valid properties or conjectures, that were justified inthe descending phase, while in the third test activity they found counter-examples for their conjectures, eliminating the need for justifications in thedescending phase.

5. CONCLUSIONS

In this paper we have reported a part of a research whose main objectivewas to analyze the variety of students’ justifications when solving proofproblems in a Cabri-Géomètre environment. To analyze students’ answers,we have defined a framework which integrates and expands different previ-ous partial approaches: The types of justifications described by Bell (1976aand b), Balacheff (1988a and b), and Harel and Sowder (1996), and thecharacterization of the shift from an empirical work (ascending phase) to adeductive work (descending phase) described by Arzarello et al. (1998a).From the analysis of results of the two case studies made in section 4, wecan formulate some conclusions:

– The types of justifications and the phases in the process of producingjustifications are complementary elements and allow us to make a de-tailed analysis of solutions to proof problems: Both product (types ofjustifications) and process (phases of solution) are important to knowstudents’ reasoning while solving proof problems, their strategies and(in)coherences among different moments or parts of the solution.

– A DGS like Cabri may well help secondary school students under-stand the need for abstract justifications and formal proofs in math-ematics. Secondary school students cannot make a fast transition fromempirical to abstract ways of conjecture and justification. Such trans-ition is very slow, and has to be rooted on empirical methods usedby students so far. In this context, DGS lets students make empir-ical explorations before trying to produce a deductive justification, bymaking meaningful representations of problems, experimenting, andgetting immediate feedback.

– Dragging is a unique feature of DGS (of Cabri in particular) thatmakes DGS environments much more powerful than traditional paper-


and-pencil learning. Dragging lets students see as many examples asnecessary in a few seconds, and provides them with immediate feed-back that cannot be obtained from paper-and-pencil teaching. In ourteaching experiment, dragging helped students to look for properties,special cases, counter-examples, etc. that could be linked to form aconjecture or a justification. In particular, the dragging test was usedmost of the times as the criterion to accept a figure as correct.

– By stating carefully organized sequences of problems, and giving stu-dents enough time to work on them, it is possible to have studentsprogress toward more elaborated types of justifications.

– The experiment reported here lasted about 30 weeks, with two 55minute classes per week. During this time, the best students (T and P)improved the quality of their justification skills, although they alwayselaborated empirical justifications. Other students made more limitedprogress, like H and C, or even no progress at all. Therefore, second-ary school students require a considerable amount of time, devotedto experiment with Cabri, to begin to feel confident with deductivejustifications and formal proofs.

– The agreed didactical contract between teacher and pupils, in refer-ence to what kinds of answers are accepted, is an important elementto success in promoting students’ progress. In our experiment, thedidactical contract made explicit by the teacher can be summarizedas the need to organize justifications by using definitions and results(theorems) previously known and accepted by the class.

– There is progress in the ability to produce justifications or proofsonly if there is parallel learning of mathematical concepts and prop-erties related to the topic being studied (see section 4.1.1). In ourexperiment, the ‘notebook of accepted results’ turned out to be anecessary aid. It gave the students ready access to all the ‘acceptedresults’. We have observed in the case study that sometimes studentsfailed to solve a problem because they did not remember a necessarygeometrical property.

A weakness of the research reported in this paper is that it is based on twocase studies of pairs of students, so only a limited variety of justificationshas been obtained. Research with more students would be necessary to geta wider variety of solutions of problems and confirm the validity of theframework defined here.

Our study is just one piece of a research agenda on the teaching andlearning of mathematical proof in DGS environments. There is still muchwork to do. We can mention some points in this agenda that still needresearch:


– There are many research studies based on students in intermediatesecondary school grades, but studies based on students in the loweror higher secondary school grades, or even university students, areinsufficient.

– There is a lack of research about the transfer of justification know-ledge and skills using DGS environments when students return to thetraditional context of teaching mathematics based on blackboard andtextbook.

– The types of justifications we have defined are not totally ordered.Nonetheless, it is useful to know if there are some paths in the de-velopment of students’ ability of justification. If existence of suchpaths is confirmed, it would be interesting to know about a possibleinfluence of DGS environments on such paths.

NOTES

1. The term proof problem (or ‘problem to prove’ according to Polya 1981, p. 1–119)refers to a kind of problem where students are asked to provide a justification for anassertion. This assertion may be explicit in the statement of the problem or may beinduced by students as the first part of the solution of the problem.

2. Laborde and Capponi (1994) used the term ‘Cabri-drawing’ to differentiate a drawingon the screen from a drawing on a sheet of paper. A Cabri-drawing is usually dynamic,but a drawing on paper is static. This differentiation is not relevant to our research;thus we do not use such term in this paper.

3. The students called them ‘opposite triangles’ because they had opposite angles in M.Opposite angles are named vertical angles in some countries.


Annex IContent of the teaching unit

Nr Tp Content of activity Accepted results

1 – Reminder of use of Cabri. —–

2 C Discover properties of the perpendicular The perp. bisector is perpendicular to

bisector. the segment and cuts it in the midpoint.

3 C Discover properties of points of the Points of the perpend. bisector are equi-

perpendicular bisector. distant from the ends of the segment.

4 C Discover properties of the angle The angle bisector divides the angle into

bisector. two congruent angles.

5 C Discover properties of points of the Points of the angle bisector are equi-

angle bisector. distant from the sides of the angle.

6 – Reminder of classifications of triangles. Classifications of triangles.

7 C Use of macro ‘compass’. Construction Method of construction of triangles

of a triangle given 3 segments (sides). given 3 sides.

8 C Discover properties of perpendicular The circumcentre and its characteristic

bisectors of a triangle. property.

9 C Discover properties of angle bisectors of The incentre and its characteristic

a triangle. property.

10 C Discover properties of altitudes of a The orthocentre. Relationship among

triangle. congruence of altitudes and types of

triangles (sides).

11 C Discover properties of medians of a The centroid and its characteristic

triangle. property.

12 CJ When do the 4 perp. bisectors of a —–

quadrilateral meet in a single point?

13 – Remainder of ways of construction of Conditions for congruence of triangles.

triangles. Uniqueness of the result.

14 C Study positions of straight lines on a Congruence of opposite angles.

plane. Angles between 2 lines.

15 CJ Angles between 2 lines in a plane. Two linear angles are supplementary.

16 CJ Angles created by 2 parallel and a Congruence of angles: Corresponding,

transversal lines. alternate exterior, alternate interior, etc.

17 CJ Sum of the interior angles of a triangle. The interior angles add up to 180◦.

18 CJ Discover properties of external angles of Relationship between interior and

a triangle. external angles. Sum of the external

angles of a triangle.

19 CJ Discover properties of isosceles Properties of the vertex angle bisector in

triangles. an isosceles triangle.

20 CJ Given AB//CD and AB=AC, is CB the —–

angle bisector of � ACD?

21 J Study angles created by 2 pairs of Relationship between different angles.

parallel lines. Study diagonals of a Characterization of diagonals of paral-

parallelogram. lelogram, rectangle, rhombus, square.


Annex IContent of the teaching unit

Nr Tp Content of activity Accepted results

22 C Discover characteristics of each class of Definition and classification of paral-

parallelogr. Relationship among classes. lelogram, rectangle, rhombus, square.

23 J Given a parallelogr. ABCD, its diagonal —–

AC, a point P in AC, and segments

NQ//AB and MR//AD meeting in P, do

NPRD and MPQB have the same area?

24 CJ Discover properties of trapeziums. Opposite angles in an isosceles

trapezium are supplementary.

25 J Discover properties of kites. —–

26 CJ Discover properties of midpoints of —–

sides of a triangle.

27 CJ Discover properties of midpoints of Varignon’s theorem.

sides of a quadrilateral.

28 J Discover properties of similarity. Some applications of Thales theorem.

29 CJ Relationships among central and Central angle = 2 × angle inscribed in

inscribed angles in a circle. the same arc. Any angle inscribed in a

semicircle is a right angle.

30 CJ Definitions of tangent and secant of a A tangent to a circle is perpendicular to

circle. Given a circle and a point P the radius drawn to the tangency point.

exterior to the circle, construct the

tangents to the circle passing through P.

Types of activities: (C) asks only for a conjecture, (J) asks for a justification of a given conjec-

ture, (CJ) asks for a conjecture and a justification of it. Bold numbers: The three ‘test activities’.

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Dpto. de Didáctica de la Matemática, Universidad de Valencia,Apartado 22045,46071 – Valencia (Spain),Phone: 34-963864486,Fax: 34-963864486,E-mail: [email protected]

NURIT HADAS, RINA HERSHKOWITZ and BARUCH B. SCHWARZ

THE ROLE OF CONTRADICTION AND UNCERTAINTY INPROMOTING THE NEED TO PROVE IN DYNAMIC GEOMETRY

ENVIRONMENTS

ABSTRACT. In many geometrical problems, students can feel that the universality of aconjectured attribute of a figure is validated by their action in a dynamic geometry envir-onment. In contrast, students generally do not feel that deductive explanations strengthentheir conviction that a geometrical figure has a given attribute. In order to cope with stu-dents’ conviction based on empirical experience only and to create a need for deductiveexplanations, we developed a collection of innovative activities intended to cause surpriseand uncertainty. In this paper we describe two activities, that led students to contradic-tions between conjectures and findings. We analyze the conjectures, working methods, andexplanations given by the students when faced with the contradictions that arose.

KEY WORDS: deduction, design processes, explanations’ categories, proving in geometry

INTRODUCTION

For generations, proofs have been considered as tools for verifying math-ematical statements and showing their universality. Leibniz believed thata mathematical proof is a universal symbolic script, which allows oneto distinguish clearly between fact and fiction, truth and falsity (Hanna,1990).

In consequence the two classical reasons for teaching proofs were, (a)to teach deductive reasoning as part of human culture, and (b) to serve asa vehicle for verifying and showing the universality of mathematical state-ments. According to this cultural perspective of proof, the addressee for‘showing’ is unspecified, and the function of proof is to fulfill the internal‘demands of mathematics’, not the psychological needs of any learner.The goal for the teaching of proof is to establish the students’ normsof mathematical knowledge: experimenting, visualizing, measuring, in-ductive reasoning and checking examples are not considered as normativemeans for this purpose.

Recently there has been a change in this approach for three reasons, thefailure to teach proofs, the recognition that the activity of proving mustcomply with the motives of the learner, and the appearance of dynamictools.


128 NURIT HADAS ET AL.

The failure to teach proofs

The failure to teach mathematical proof appears to be universal. Senk(1985) found that only 30% of the students in full year geometry coursesthat teach proofs (in the U.S.A.) reached 75% mastery in proving. In ad-dition, even those students that succeeded to function in the proving ritualwere not always aware of its meaning. They rarely saw the point of prov-ing, and/or the need to prove, especially when the statement to be provedwas given as a ready-made fact without any discovery by the learners.Balacheff (1991) claims that if students do not engage in proving pro-cesses, it is not so much because they are not able to do so, but ratherbecause they do not see any reason or feel any need for it. High-schoolstudents, even in advanced mathematics and science classes, do not real-ize that a formal proof confers universal validity to a statement. A largepercentage of students believes that, even after they have proved, checkingmore examples is desirable (Fischbein and Kedem, 1982; Vinner, 1983).Many students do not distinguish between evidence and deductive proofas a way of knowing that a geometrical statement is true (Chazan, 1993).Similarly, Schoenfeld (1986) claimed that after a full year geometry coursein college, most students do not see the point of using deductive reasoningin geometric constructions, and that they are still naive empiricists whoseapproach to constructions is an empirical ‘guess and test’ loop. They pro-duce proofs because the teacher demands them, not because they recognizethat they are necessary in their practice (Balacheff, 1988).

The goals of teaching proofs

As mentioned above, for mathematicians, proofs play an essential role inestablishing the validity of a statement and in shedding light on the reasonsthat support that statement. Lately mathematics educators have extendedconsiderably the goals that must be fulfilled when teaching proofs. Forexample de Villiers (1990, 1999 p. 5) lists six functions of proof:

• Verification (concerned with the truth of a statement),• Explanation (providing insight into why it is true),• Systematization (the organization of various results into a deductive system

of axioms, major concepts and theorems),• Discovery (the discovery or invention of new results),• Communication (the transmission of mathematical knowledge),• Intellectual challenge (the self-realization/fulfillment derived from construct-

ing a proof).

SETTLING CONTRADICTION BY PROVING 129

It seems that the first two functions mentioned by de Villiers, are relatedto the main roles of proof in mathematics discussed above: – verificationdirectly relates to validation and explanation is synonymous with sheddinglight. An analysis of teaching materials indicates that these functions oftenremain hidden in textbooks. The list presents new perspectives on proving,both from the point of view of the professional mathematician (as shownabove) and the experienced teacher. De Villiers considers these functionsas desirable, although he is aware that they are not fulfilled in class.

Balacheff (1988, 1991) identifies roles of proof, which, in a sense, aresimilar to the communication function mentioned by de Villiers. Like deVillliers, Balacheff also notes that such a desirable function is not fulfilledin class and suggests that classroom activities in which students becomeaware of such aspects of proofs be created.

Hersh (1993) and Hanna (1995) assert that the main function of proofin the classroom should be to promote understanding by explaining (whichis also in de Villiers’ list).

The existence of Dynamic Geometry (DG) environments

The appearance of dynamic geometry environments raised a question con-cerning the role of proof in the curriculum, since conviction can be ob-tained quickly and relatively easily by dragging. Dragging a geometricalobject enables students to check the invariance of a conjectured attribute ona whole class of objects. Such an operation leads students to be convincedof the truth of the conjectured attribute. Therefore DG environments mayprevent students from understanding the need and function of proof. Thesesuggestions are corroborated by Yerushalmy, Chazan and Gordon (1993),who have investigated how students function in open inquiry activities inDG environments, that support experimentation, conjecturing, checkinginvariant properties of a figure, and thus lead to conviction. De Villiers(1997, 1998) illustrated how one can enrich investigations in DG environ-ments by asking ‘what if’ questions, to lead to generalizations and discov-eries. He claimed that the search for proof then becomes an intellectualchallenge, stemming from the need to understand why the conclusion istrue, and not an epistemological exercise in trying to establish truth. Inaddition Goldenberg, Cuoco, and Mark (1998) stated that:

A proof, especially for beginners, might need to be motivated by the uncertaintiesthat remain without the proof, or by a need for an explanation of why a phe-nomenon occurs. Proof of the too obvious would likely feel ritualistic and empty(p. 6).

They concluded that DG environments may provide opportunities for thecreation of uncertainties, leading students to seek for explanations.


Movshovitz-Hadar (1988) identified the role of surprise in supportingthe teaching of proofs. Similarly, Dreyfus and Hadas (1996) argued thatstudents’ appreciation of the possible functions of proof can be achieved byactivities in various learning environments (with and without DG software)in which the empirical investigations lead to unexpected, surprising situ-ations. In other studies (Hadas and Hershkowitz, 1998, 1999) we demon-strated how appropriate activities in DG environments can be carefullydesigned to create situations of contradiction followed by surprise or byuncertainty, to lead students to seek for explanations.

In summary, the findings concerning the failure to teach proofs, therecognition of the multiple aspects of proving, and the existence of DGtools, lead naturally to the design of investigate situations in which DGtools may foster these multiple aspects.

THE STUDY AND ITS FRAMEWORK

In this paper we describe a study involving two activities designed topromote several aspects of proving in a dynamic geometry environment(Geometry Inventor, 1994). Each activity consists of a sequence of tasks,which together create a global coherent activity. We made use of 6 possiblemediating functions of the DG environment:

• To serve as learning environment for raising conjectures.• To refute (or confirm) an initial conjecture (formulated with or without

the DG tool), and in cases of refutation, to create a contradictionbetween expected results and actual findings.

• To ‘push’ students from one conjecture to a second one in cases wherethe conjecture is neither refuted nor confirmed (i.e., in situations ofuncertainty, see discussion in section 4).

• To lead students to be convinced that a conclusion is right, based oninductive trials.

• To enable the construction of an existence example.• To provide additional sources of explanation.

The first activity was designed to create a surprise resulting from a dialecticprocess stemming from contradiction between students’ hypotheses andtheir findings and leading to the need for explanation. The setting fromwhich the surprise arose left the contradiction unresolved. The need toexplain thus emerges quite naturally.

In the second activity, contradiction also occurred, but led to uncertaintyin addition to surprise, since students were not sure whether a specific geo-metrical setting can exist. This type of activity has already been mentioned


by Movshovitz-Hadar (1988), and is in correspondence with a situation ofcreating surprises by the ‘existence of the unexpected’ (p. 35). In orderto overcome the uncertainty, students either find or construct an example,or explain why such an example is unconceivable. In both cases, carefuldesign triggers students to base their actions on deductive considerations.The environment serves as a laboratory for overcoming uncertainty.

Each of the two activities was designed through a process of design-research-design as an outcome of semi-structured interviews (Hadas andHershkowitz, 1998; Hershkowitz et al., in press). The version presentedin this paper is the end product of this process. In each case, we firstbring the activity itself, then analyze the role of the various tasks in theactivity, and present some global findings on students’ actions, and finallywe describe both qualitatively and quantitatively the various categories ofstudent explanations.

The categorization of students’ explanations is similar to ‘proof schemes’of college students made by Harel and Sowder (1998). Harel and Sowderuse three main categories (each of which is divided into subcategories): 1.‘External conviction proof schemes’ where doubts are removed by ritualproof, by symbolic form arguments without meaning, by authority, or bymemorizing statements. 2. ‘Empirical proof schemes’ where conjecturesare validated or refuted by physical experience, perceptual or inductivearguments. 3. ‘Analytic proof schemes’ where the statement is validatedby logical deduction which can be transformational or axiomatic.

Our approach is ‘bottom-up’, as we aimed to classify students’ explan-ations in cases where contradiction and uncertainty situations are createdand investigated by students with the mediation of the DG tool. Hence fivecategories were defined by which we were able to describe qualitatively aswell as quantitatively students responses.

The two activities in this research are part of a course that covers theofficial syllabus in Israel, according to which students learn geometry for2 years (from mid Grade 8 to mid Grade 10), two periods per-week, withemphasis on proving. The students in this study were from Grade 8 forthe first activity and from Grade 10 for the second activity. Both researchpopulations belong to the upper 60% of the entire population.

Each activity was used first as a research tool for semi-structured in-terviews with pairs of students, and second for written report of pairsof students in a few experimental classrooms. The interviews enabled usto trace the changes and development in students’ conjectures and ex-planation processes in contradiction and uncertainty situations. All writtenreports of students working in pairs (from classes and interviews) werecollected and analyzed according to students’ conjectures and explana-


tions. The results of this analysis provided a quantitative dimension thatclarified to what extent students faced contradictions, to obtain a variety ofexplanations and to establish a categorization.

THE ROLE OF CONTRADICTION IN ‘THE SUM OF EXTERIOR ANGLES’ACTIVITY

The ‘sum of exterior angles’ activity belongs to the beginning of the geo-metry course. Students had already studied an introductory chapter aboutparallel lines including the sum of angles in a triangle. They were alsofamiliar with the dynamic geometry software.

In many traditional geometry textbooks students are given a task similarto the following:

a. Prove that the sum of the interior angles of a polygon is 180◦(n-2), where nis the number of sides.

b. Prove that the sum of the exterior angles of a polygon is 360◦.

A more open approach using a dynamic geometry environment gives stu-dents the opportunity to discover these attributes before being invited toprove (Yerushalmy and Chazan, 1990). We designed an activity intended tocreate a contradiction between plausible student conjectures about the sumof the exterior angles as the number of polygon sides increases, and theresults obtained by the measuring operation done in the DG environment.We aimed to investigate the effect of the activity on students’ explanations,and on their feeling for the necessity to prove. The wording of the activityis as follows:

Task A: Measure (with the software) the sum of the interior angles in polygonsas the number of sides increases. Generalize, and explain your conclusion.

Task B: Measure (with the software) the sum of the exterior angles of a quadri-lateral.Hypothesize the sum of the exterior angles for polygons as the number ofsides increases.Check your hypothesis by measuring and explain what you found.

Comments: 1. In both tasks, students were first led to check (by dragging)the sums of the interior/exterior angles for polygons with the same numberof sides, and only then to change the number of sides. 2. The DG softwaremeasures only angles smaller than 180◦. Therefore we confine this activityto convex polygons.


Both tasks were deliberately posed in open form to increase the contradic-tion between the conjectured and the invariant property of the sum of theexterior angles, and thus the need to explain the results.

The goal of Task A was to familiarize students with the situation and toprovide an opportunity to generalize the increase in the sum of the interiorangles as the number of sides increases, without necessarily relying on al-gebraic considerations. After generalizing, students were asked to explaintheir generalization. The generalization and explanations in Task A mayencourage students to conjecture that the sum of the external angles is alsoincreasing. Consequently, the contradiction that students may face in TaskB, is likely to become stronger.

The activity was first given in the form of a worksheet to 4 pairs ofstudents (from grade 8) in a semi-structured interview. The interviewerencouraged students to make conjectures before measuring, to check theirhypotheses by measuring, and then to explain verbally their universality.In addition, each pair of students reported their hypotheses and explan-ations on the worksheet. The interviews were videotaped, analyzed andinterpreted.

In a second stage, the activity was given to 82 students in three 8grade classes, with students working in pairs. As a result we collected andanalyzed 45 reports (4 from the interviews and 41 from classrooms). Inthe following we analyze the tasks, reports and interviews, and describefindings and the types of explanations given by students in interviews andin class.

Activity analysis and findings

Task A:(i) Students in 32 reports generalized from working with the DG softwarethat the sum of the interior angles increases by 180◦ when the number ofsides increases by 1. (ii) In 18 reports students expressed their generaliz-ation algebraically: 180◦ (n-2), 180◦(n-4) + 360◦, or 180◦n – 360◦. (iii) 5pairs gave both kinds of generalization.

Following the request to substantiate their generalizations, students pro-duced various explanations. Most based their explanations by generalizingtheir measurments, or, by adding a triangle when the number of sidesincreases by one (Figure 1).

Task B:In 37 of the 49 responses (the 8 students in the 4 interviewed pairs whohypothesized individually, and the 41 written responses in the classrooms),students hypothesized that the sum of the exterior angles increases with the


Figure 1. Students’ drawings in various explanations for Task A, Activity 1.

number of sides. Another 10, that were all from the same class, hypothes-ized that the sum is fixed. From the information we had from the teacherof this class, students knew this fact from previous work. Two girls, in aninterview, did not make any hypothesis.

Task A pushed many students to generate the hypothesis that the sum ofthe exterior angles increases. The degree of confidence that the hypothesisis true became quite strong as reflected in the following episode.

In the course of a discussion, Jonathan claimed that the sum wouldalways remain 360◦. Galia asked him if he thought that when the numberof sides increases beyond 100, the sum would still remain 360◦. Jonathanhesitated and then agreed that the sum would increase. This episode indic-ates that the intuitive belief that the sum increases with increasing numberof sides, is quite strong even when the student has some knowledge aboutthe true statement. This belief may produce contradiction and surprisewhile checking this conjecture, and the contradiction may trigger a needfor explanation.

Types of explanations

Fifty explanations relating to Task B were given by the students (in theinterviews students gave more than one explanation), to refute (or confirm)the initial conjecture. The explanations were classified by the authors, intothe following five categories, and two other experienced teachers validatedthis classification (as Task A does not lead to contradictions, we did notinclude the explanations given by students to this task in the analysis.)


1. No explanation categorySeventeen of the 50 responses (34%) belonged to this category. It includedresponses without any argument, responses that were mainly tautological(i.e., in which students rephrased the result of their measuring and addedsome ‘supporting’ words as in Example 1), and responses whose argu-ments relied on external authority.

Example 1:The sum of exterior angles in every polygon, either pentagon or 7 sided polygon,is always 360◦.

2. Inductive explanation categoryIn two of the explanations (4%), inferences, not based on general deduct-ive arguments, were drawn from features of one or several examples. Forexample, in one of the interviews, Noa and Inbar explained that the sum ofthe exterior angles is always 360◦ by checking numerically as follows:

Example 2:

Noa: We will multiply 180◦ by the number of sides and we will subtract theinterior [sum of angles] we found [in Task A].

I: And what we will get from it? It will increase or decrease? By how much?Noa+Inbar: We don’t know.

They started to write down a table:

Interiors + Exteriors Interiors Exteriors

180*4 – 360 = 360◦

180*5 – 540 = 360◦

180*6 – 720 = 360◦

Noa: The sum is always 360◦, and we don’t have to continue to check. But, westill don’t know why the sum doesn’t change.

Although Noa and Inbar elaborated a global process for computing theexterior sum and were convinced of its universality, they did not considertheir inductive process as an explanation of why it is true.


3. Partial deductive explanation categoryThere were six explanations (12%) in this category.

Example 3:A pair of classroom students who concluded in Task A that the sum of theinterior angles is 180◦(n-2), explained in their written response to Task B:

One should take off the sum of the interior angles from the sum of the adjacent[interior and exterior], and then one gets 360◦.

This is a deductive type of explanation in which the 2 computations, 180◦nand 180◦n – 180◦(n-2) are missing.

4. Visual-variations explanation categoryIn this category we included explanations that make use of visual variation,which might be a result of imagery or the dragging action. As such, thiscategory is particular to explanations in DG learning environments. Thesixteen explanations (32%) in this category include variations based onvisual imagery, described verbally and by drawings.

Example 4:A pair of students noted a visual attribute of the sum of the exterior angles:

There is a whole turn around the polygon, therefore the sum is 360◦.

And they drew Figure 2.

Figure 2. A ‘whole turn around a polygon’ as an explanation for Task B, Activity 1.

Their use of imagery for expressing a global visual property of poly-gons, which is independent of the number of sides, is clear. Harel andSowder (1998) call such arguments transformational analytic proof schemes.

Note: This example is included in category 3 also, because the students basedtheir explanation on the statement that a complete turn is 360◦.

Example 5:A typical response in this category is the following:


When the number of sides increases, we have more angles. But they are smaller,because the internal angles become bigger.

The Inventor DG software, enables students to choose a polygon with agiven number of sides. The polygon appears in its regular form which canbe ‘messed’ by dragging. Hence, students who choose polygons (createdby rays) sequentially with increasing number of sides, see the added anglesimultaneously with the decreasing measure of all the exterior angles.

5. Deductive explanation categoryIn this category there were nine explanations (18%). They consist of a com-plete chain of logical arguments as illustrated by the following examples.

Example 6:In their interview, Tomer and Adir, hypothesized that the sum of the ex-terior angles will be always equal to the sum of the interior angles (asin quadrilaterals). After measuring with the DG tool, they concluded thatthe sum of the exterior angles is constant and equal to 360◦, relying oncompensation between the sums of the interior and exterior angles. Theythen checked numerically a sequence of polygons with increasing numberof sides, like the two girls in Example 2. They calculated the sum forpolygons of 4, 5, and 6 sides and then ‘jumped’ to 12 sided polygons.They felt then that they had reached a dead end:

Adir: 12 sides it is 12 multiplied by 180, minus the sum inside which is. . .

They did not have the formula nor numerical information about the sum ofthe interior angles of 12-sided polygons.

Adir: I can say that the sum of all pairs of the adjacent angles [the interior andexterior angles together] minus the entire interiors, equals the sum of theentire exteriors.

I: O.K. But how can you be sure that when you subtract the sum of the interiorangles, that is changing all the time, the sum of the exterior angles will stay360◦?

Adir: I can explain this to you: to the sum of all the interiors 180◦ is added, and180◦ is added to the sum of all the adjacent. And if I subtract the sum of theinterior. . .

Tomer: Then this is coming back. . .

Adir: It compensates the 180◦ back, it remains the same. Let us write it downTomer.

(Adir was writing what Tomer summarizes):


Tomer: When we add one side to a polygon, the sum of the interior angles in-creases by 180◦, and the sum of the adjacent angles increases by 180◦.Therefore when we subtract the sum of the interior from the adjacent, weget the same number.

Adir: And it starts at 360◦ because it is like that in a triangle, and this is thesmallest polygon one can get.

In addition to explaining why the sum of the exterior angles in polygons isconstant, Adir felt the need to complete the explanation why this constantsum is equal to 360◦. We can conclude that the students felt that theirexplanation is adequate, from their need to summarize it by writing. It isworth noting that because this example is the record of an interview, weare able to see the process of construction of the justification.

Additional pairs of students used the explanation element they pro-duced in Task A – the triangle they built (See Figure 1.a) – to explainwhy the sum of the exterior angles is fixed. For example: Inbal and Limorin the following episode.

Example 7:

Inbal: If we add a triangle, it is like the sum of the angles is not. . . I am not sureabout it. . .

Inbal then drew Figure 3 to support her process of explanation.

Figure 3. A drawing for explanation based on adding a triangle for Task B, Activity 1.

Inbal: To continue?I: Yes, Yes. . .

Inbal: This is what was added (γ ), and this is what was reduced (Inbal is pointingat α & β).

Limor: This was outside before (pointing at α & β). [α is opposite to a part froman external angle].

Inbal: Because this is ahh. . .I know why. . .the sum of the exterior angles. . . This(pointing at γ ) is equal to these two (pointing at α & β) it is equal for bothof them.

Limor and Inbal: Yes, yes it is true.


Both girls constructed their explanation while talking and drawing cooper-atively. From their last common sentence above, it seems that they becameconfident that their explanation was correct. It is worth noting that althoughthe girls used a special drawing to exemplify their explanation, in Task Athey talked about adding a triangle as general way of adding a side. Thuswe consider their explanation as also general.

Summarizing, as intended in the design of the activity: (i) Many stu-dents in Task B faced the contradiction between their hypothesis and theresults; (ii) In spite of being at the beginning of the course on Euclideangeometry, nine of the fifty explanations were complete deductive explan-ations and eight more used partial deductive or inductive explanations;(iii) The generalizations and the explanations students produced in TaskA, served as artifacts in their explanations in Task B (see examples 6 and7).

ROLE OF UNCERTAINTY, IN THE ‘CONGRUENT TRIANGLES’ ACTIVITY

The ‘congruent triangles’ activity was designed for the last months of thegeometry course (Grades 9 and 10). Students were asked to investigatesufficient conditions (equality of sides and angles) for congruence in a situ-ation of uncertainty. The uncertainty emerged from situations in which theconstruction was possible, but was opposed to students’ intuitive conjec-tures or alternatively in situations in which the construction was impossibleand some students conjectured that it was possible. Students at that stagewere already familiar with congruence and similarity and made use of therelevant theorems.

The activity

In this activity, we will investigate if and when, two triangles having several equalelements, are congruent.

Part 1: Task 1a. Given a dynamic �ABC, construct another �DEF having oneangle and one adjacent side equal to one angle and adjacent side of �ABC.Drag the vertices of both triangles and investigate the variance of the tri-angles and how they relate one to another.

Task 1b. Given a dynamic �ABC, construct another �DEF having twoangles and the included side equal to two angles and the included side of�ABC. Drag the vertices of both triangles and investigate the variance ofthe triangles and how they relate one to another.

Task 2. Is it possible to construct a triangle with one side and two anglesequal to those of a dynamic �ABC, but not congruent to �ABC? If it ispossible, construct such a triangle; otherwise explain why.


Part 2: Task 3a. How many equal sides and angles are there in the non-congruenttriangles constructed in task 2?

Task 3b. Is it possible to construct two non-congruent triangles, with fiveequal elements (sides and angles)? Generate an hypothesis.

Task 4. Is it possible to construct two non-congruent triangles with six equalelements? If yes, construct two such triangles, otherwise explain why.

Part 3: Task 5a. Let’s suppose that we already have 2 non-congruent triangleswith five equal elements, specify the equal elements.

Task 5b. Try to construct two non-congruent triangles, with five equal ele-ments.

This activity was first given to 2 pairs of Grade 10 students in semi-structuredinterviews, which were videotaped and analyzed (for more details and ana-lysis of the interviews, see Hadas and Hershkowitz, 1999). In the secondstage the activity was given in a Grade 10 classroom, where students workedin pairs. As a result 12 written reports (10 from the class and 2 from theinterviews) were collected and analyzed.

In the following we first analyze the various tasks in the light of stu-dents’ actions and conjectures and then discuss the various types of ex-planation qualitatively and quantitatively.

Activity analysis and findings

• Task 1 was aimed to invite students to grasp the concept of congru-ence as a possible invariant under dragging. In Task 1a, both trianglescan be changed by dragging their vertices and they may or may notbe congruent, because they have only one side and one angle equal.On the other hand, in Task 1b, where there is an additional equalangle, the second triangle (�DEF) cannot be changed by dragging itsvertices, and when one changes �ABC by dragging, �DEF changesaccordingly and remains congruent to it. This task prepares studentsfor Task 2 when they realize that congruence may be an invariantattribute under dragging.

• In Task 2, four pairs of students constructed two non-congruent tri-angles satisfying the required conditions. The other eight pairs claimedthat this construction was impossible because of the congruence the-orem of one side and two angles. These eight pairs then received Task2*.

In Figure 4, side AC = DE and <A = <D. Examine all possible ways of‘copying’ an additional angle of �ABC, in order to complete the secondtriangle. Is the completed triangle congruent to �ABC for all possibilities?


Figure 4. The drawing for Task 2*, Activity 2.

This task ‘pushed’ these eight pairs of students to construct non-congruent triangles with one side and two angles equal. The DG soft-ware enables students to try various ways of ‘copying’ �ABC anglesto different vertices in �DEF, and to get feedback. They then realizethat when the ‘copying’ does not preserve the correspondence, thetwo triangles are not necessarily congruent. This means that eventhough dragging the vertices of �ABC changes �DEF, the two tri-angles are generally not congruent.

• When in Task 3a, students were asked to identify the equal elements,eight pairs were surprised to discover four equal elements in the twonon-congruent triangles. The remaining four pairs claimed that thethree elements used in the construction are the only equal elements. Itseems that the belief that three equal elements are sufficient to guar-antee congruence, causes students to be surprised or prevents themfrom concluding the equality of the forth element in the second case.

• While solving Task 3b, the pair of students in the first interview con-sidered the possibility of constructing a triangle non-congruent to�ABC, with six elements equal to elements in �ABC. We conse-quently decided to insert Task 4, and thus obtained three parts: aconjecture (Task 3b), a discussion of the case of six equal elements(Task 4), and finally the construction of a triangle with five elementsequal to elements of �ABC, and yet non-congruent to it (Task 5b).

• The two pairs in the interviews conjectured in Tasks 3b and 4 that itis possible to construct non-congruent triangles with five or six equalelements. The discourse in the interview is described in Hadas andHershkowitz (1999). The ten pairs from the classroom conjecturedfirst that it is impossible to construct non-congruent triangles withfive or six equal elements, based on their belief that three equal ele-


ments guarantee congruence although they had already constructednon-congruent triangles with four equal elements. Following a wholeclass discussion, they also felt a need for an explanation. Hence wecan conclude that all 12 pairs faced a situation in which they had toexplain why it is impossible to construct non-congruent triangles withsix equal elements.

• Task 5 was designed to challenge students’ conjecture that triangleswith five equal elements must be congruent. This task comes afterTask 2 where students were surprised to discover that they can con-struct two non-congruent triangles with four equal elements, and afterTask 4, in which the students explained why it is impossible to con-struct non-congruent triangles with six equal elements. This sequencemade students uncertain about the problem of five equal elements.This uncertainty encouraged students to link considerations to oneof the two alternative reasons they had discovered previously namelyrelying on a congruence theorem or on lack of correspondence. Onlyone pair in the interview started directly with a construction (de-scribed in Hadas and Hershkowitz, 1999). All the additional pairsreceived Task 5*, which was intended to trigger students to use simil-arity consideration.

In �ABC (see Figure 5) one side is 6 units long and a second is 9 units long.Design a construction of �DEF with sides of 6 and 9 units also and anglesequal to the angles in �ABC, but which is not congruent to �ABC (Clue:Calculate first the length of the third side in each triangle).

Figure 5. The drawing for Task 5*, Activity 2.

Activity characteristicsFirst, in this problem the students do not receive any confirmation or refuta-tion of their initial conjectures. Thus, they are uncertain about the existenceor the non-existence of the required construction. They therefore were en-gaged in a situation of uncertainty. Secondly, as in all other constructionproblems in such an environment, the computerized tool elicited empir-ical actions followed by prompt feedback, but deductive considerations


were needed for complete success. More generally, students’ constructionprocesses depend here on their ability to analyze the problem deduct-ively, leading to the elaboration of an existence proof. In this case theconstruction itself is a kind of deductive proof – ‘proving by doing’.

Types of students’ explanations

There were a total of 112 responses in the 12 written reports on the all fivetasks of this activity. Eighty-five of them related to tasks 2 to 5 in whichstudents worked in situations of uncertainty. The analysis below relates tothese 85 explanations (the preparatory Task 1 does not lead to contradictionand was not included in the categorization of explanations.)

In spite of the differences between the tasks in this activity and theprevious one (the sum of exterior angles in a polygon), and hence thedifferences in the explanations, we found that we could use the same cat-egories.

1. No explanation categoryThere were 23 (27%) responses in this category. For example:

Two triangles with 5 equal elements must be congruent.

2. Inductive explanation categoryThree explanations (4%) fell in this category.

Example 8:One pair, while explaining why there are no congruent triangles with fiveequal elements, drew Figure 6 on their worksheet and wrote:

Figure 6. A single drawing as an explanation for the impossibility to accomplish theconstruction in Task 5*, Activity 2.


We tried, and something like that (Figure 6) came out. We can see that if onewants the triangles not to be congruent, one of the two sides we choose will neverbe 9. [They meant, that EM is never 9].

The two students relied on manipulations on one example to explain theirwrong conclusion.

3. Partial deductive explanation categoryIn nine (11%) explanations students constructed a partial deductive chainof arguments, which mostly led to a wrong conclusion.

Example 9:One pair from the whole class population, explained in Task 3b:

If the 3 sides and 2 angles are equal it is impossible [to construct non-congruenttriangles]. If 2 sides and 3 angles are equal it is also impossible, because then thetriangles are similar, and it does not exist without correspondence.

This pair of students did not take into consideration that similar trianglescould have two equal (not corresponding) sides and be non-congruent. Inspite of the fact that their conclusion was wrong the arguments which tietogether the different claims were deductive. The wrong conclusion seemsto indicate their difficulty in accepting a situation, which contradicts the in-tuition based on previous knowledge in which three elements are sufficientfor congruence.

Example 10:In Task 4, Nadav and Gili (in an interview) discussed the issue of six equalelements in non-congruent triangles. Nadav who was surprised to succeed(in Task 2) to construct non-congruent triangles with four equal elements,argued that it was possible to construct non-congruent triangles with sixequal elements and tried to apply the same strategy again – drawing equalelements but not in correspondence. He drew Figure 7 on a piece of paper,and asked the interviewer whether to start with the construction in theDG environment. It is interesting to note that yet he sketched congruenttriangles.

Figure 7. Nadav sketch for Task 4, Activity 2.


Later the two students concluded deductively that this construction wasimpossible.

4. Visual-variations explanation categoryWorking in a DG environment creates opportunities for explanations arisingfrom visual variation induced by dragging. This kind of visual variationwas absent in the explanations of the first activity. We classified two ex-planations in this category (2%); one of them is also included in the de-ductive category. It is worthwhile noting that in Task 1, seven pairs usedthis kind of explanation to explain why one side and one angle are notsufficient for congruence, but one side and two corresponding angles aresufficient.

Example 11:Following their success in constructing non-congruent triangles with fourequal elements and their correct inference that one cannot construct twonon-congruent triangles with six equal elements, (see example 10), Nadavand Gili, tried to solve the uncertainty in the case of five equal elements.They erased the ‘equal signs’ on BC and DE in their paper and pencildrawing (see Figure 7) in order to plan their construction. Then they startedto drag the vertices of �ABC (with four elements equal to four elements in�DEF) already constructed in Task 2, and tried to get an additional equalside through a sequence of dragging and measuring actions.

Example 12:In Task 2, a pair of students in an interview constructed the second triangle(see Figure 8) where: DE=AC; <D=<A; and <F=<B, and then draggedvertices in order to try to obtain non-congruent triangles, until they realizedthat it was impossible. They then explained:

As a matter of fact, It was not possible to drag vertices B or F, in order to makeside DF unequal to AB.

5. Deductive explanation categoryForty-eight explanations (56%) fell in this category. All the 12 pairs con-structed non-congruent triangles with one side and two angles equal, eitheras a response to Task 2 or to Task 2*. The deduction here is expressed bythe doing itself, because without the deductive considerations of corres-pondence they would not be able to complete the task. An additional sixconstructions, to solve Task 5 or 5*, are examples of deduction by doing.The students deduced that the triangles (see Figure 5), are similar becauseof the angles’ equality, but not congruent, thus FD should be six and DE


Figure 8. Dragging vertices as an effort to get non-congruent triangles in Task 2, Activity2.

should be nine. Then they were able to calculate that the ratio betweencorresponding sides in the similar triangles DEF and ABC must be 1.5,and hence AB must be 4, and FE 13.5. Then they constructed the twotriangles with the software.

In addition to the 18 deductive explanations by doing, there were further30 typical explanations in the deduction category. For example;

Example 13:The following discussion took place after Nadav and Gili in the interview(see examples 10 and 11) calculated the ratio and the lengths of the sidesin Task 5*.

Nadav: But the angles should be equal as well, (pause) but it is O.K.I: Why are you sure that the angles should be equal?

Nadav: Because they [the triangles] are similar.

The overall goal of this activity, to push students towards deductive con-siderations in order to overcome uncertainty in geometrical situations, isexemplified in the following episode.

Example 14:Udy and Amir, a pair in the classroom, wrote in Task 2 that it is impossibleto construct non-congruent triangles with one side and 2 angles equal,because of the angle, side, angle theorem. Later in Task 2* (Figure 4) theywrote:

Angle B can be copied in two different ways, at vertex E or F. It must be copiedat E otherwise the triangles will be congruent.

In Task 3 they wrote again that it is impossible to construct non-congruenttriangles with five elements equal, and added:


In this way we will get congruency for sure, because one of the congruencetheorems should hold.

In Task 5a they wrote that the equal elements must be three angles and twosides:

because otherwise the side, side, side theorem would hold automatically . . . But,we think that even in this case the triangles will be congruent, because some othertheorem must hold.

When in Task 5*, the students succeeded in calculating the third side ineach of the two non-congruent triangles, they wrote:

In the presence of the new data that were revealed to us, we must admit that twonon-congruent triangles with 5 identical properties, do exist.

This example demonstrates how deduction by doing may solve uncertainty.

CONCLUSION

In this paper we have described two activities. In the first, students facedcontradiction and in the second they also faced uncertainty. We studiedhow these activities led many students to articulate explanations. As re-searchers, our approach is naturalistic, meaning that we classify students’explanations without relating to their correctness. As mathematics educat-ors, and as people involved in the development of mathematical thinking,we are obviously interested in fostering deductive explanations (the pre-ferred norms in the mathematics milieu). Thus we intentionally soughtto identify the frequency of deductive explanations and the conditionswhich encouraged them. However, the appearance of other explanations,especially the visual-variational explanation, often based on the (dynamic)displays or stemming from students’ imagery (which in turn may origin-ate from students’ prior experience with the DG tool), open an excitingissue. The question is whether these explanations can be considered asa normative way to prove, which obviously depends on the explanationsthemselves. This issue which deserves a special debate, stresses that normsare not only a function of what is imposed by professionals, but may beinfluenced by practices induced by new cultural tools and contexts.

As to the findings concerning the context in which deductive explana-tions arose, we saw that in the first activity, in spite of the contradiction,students were certain that the result they obtained from the tool was cor-rect. In contrast, in the second activity, we capitalized on uncertainty in ad-dition to contradiction, uncertainty that was maintained as long as the taskswere not solved. The sequence of tasks led students to oscillate betweentwo different hypotheses (the impossibility or possibility of constructing


non-congruent triangles). Students’ actions relied on deductive considera-tions, or on the inclination (based on experience from previous stages oflearning) to think that three equal elements guarantee congruence. In thissecond activity some explanations had to be expressed by doing, whichrequires the use of deductive considerations. It seems that these actionspushed them towards deductive explanations in other tasks in this activity.This may be one of the reasons for the fact that the proportion of deductiveexplanations (56%) in the second activity is much greater than in the first(18%).

The two activities exemplify a design in which learning in a DG en-vironment opens opportunities for feeling the need to prove, rather thanconsidering proving as superfluous. In addition, the design of each activity(in particular the second one) as a chain of tasks, enabled students to beinvolved and apprehend a comprehensive topic in geometry. In a sense,the second activity seemed to push students to feel the systematization (deVilliers, 1999) of the topic of congruence, – a function of proof that isseldom reached in classrooms.

In both activities, we capitalized on surprise. Surprise occurs whenthere are expectations which, do not fulfilled. The students had made aconjecture concerning the solution of the task and found a reason whythis conjecture was true. The reason was often rooted in common senseor based on previous learning. In the first activity the refutation was givenby students’ findings in the DG environment, and thus lead students tomore than accept the correctness of the new conjecture, it lead them toconstruct a new argument. In the second activity we documented and ana-lyzed refutations that evolved when the proof by doing was completed,and thus a surprise took place. We believe that in both cases students wereled to use deductive arguments, that is, to construct reasons to supportthe new conjecture or reasons that enabled them to solve the problem. Itseems that in several cases, the construction of a new argument on deduct-ive grounds was accompanied by an evaluation of the previous argument.Students related to the previous reasons to show why those reasons wereerroneous, and why the new conjecture was true. For example, studentsinitially thought that the sum of the external angles increases as the numberof sides increases. When they realized that the sum is fixed, they repairedthe argument by introducing the idea of compensation (Examples 3, 5, 6and 7).

The various explanations seem to indicate that students started to beaware to the role of explanations in general and to the role of deductiveexplanations in particular, in the construction of meaning and in increasingtheir confidence in the geometrical conclusions.


We conclude then that the design in a DG learning environment broughtproofs into the realm of student activity and argument; that is, they engagednaturally in true mathematical activity. Laborde (1995) claimed that it isimportant to design tasks for DG environments, which can only be solvedusing geometrical knowledge. And indeed in these activities, the studentsceased to be a recipients of formal proofs, but were engaged in an activityof construction and evaluation of arguments in which certainty and under-standing were at stake, and they had to use their geometrical knowledge toexplain contradictions and overcome uncertainty.

REFERENCES

Balacheff, N.: 1988, ‘A study of students’ proving processes at the junior high schoollevel’, Paper presented at the 66th Annual Meeting of the National Council of Teachersof Mathematics, U.S.A.

Balacheff, N.: 1991, ‘The benefits and limits of social interaction: the case of mathemat-ical proof’, in A.J. Bishop, S. Mellin-Olsen and J. Van Dormolen (eds.), MathematicalKnowledge: Its Growth Through Teaching, Kluwer Academic Publishers, Dordrecht,The Netherlands, pp. 175–192.

Chazan, D.: 1993, ‘Instructional implications of students’ understandings of the differencesbetween empirical verification and mathematical proof’, in J. Schwartz, M. Yerushalmyand B. Wilson (eds.), The Geometric Supposer, What is it a Case of?, Lawrence ErlbaumAssociates, Hillsdale, NJ, pp. 107–116.

De Villiers, M.: 1990, ‘The role and function of proof in mathematics’. Pythagoras 24,17–24.

De Villiers, M.: 1997, ‘The role of proof in investigative, computer-based geometry: somepersonal reflections’, in J. King and D. Schattschneider (eds.), Geometry Turned On!Dynamic Software in Learning, Teaching, and Research, The Mathematical Associationof America, Washington, DC, pp. 15–24.

De Villiers, M.: 1998, ‘An alternative approach to proof in dynamic geometry’, in R. Lehrerand D. Chazan (eds.), Designing Learning Environments for Developing Understandingof Geometry and Space?, Lawrence Erlbaum Associates, Hillsdale, NJ, pp. 369–393.

De Villiers, M.: 1999, ‘The role and function of proof’, in M. De Villiers (ed.), RethinkingProof with the Geometer’s Sketchpad, Key Curriculum Press, pp. 3–10.

Dreyfus, T. and Hadas, N.: 1996, ‘Proof as answer to the question why,’ Zentralblatt fürDidaktik der Matematik, ZDM 28(1), 1–5.

Fischbein, E. and Kedem, I.: 1982, ‘Proof and certitude in development of mathematicalthinking’, in A. Vermandel (ed.), Proceedings of the 6th PME Conference, UniversitaireInstelling Antwerpen, pp. 128–132.

Geometry Inventor: 1994, Logal Educational Software Ltd., Cambridge, MA, U.S.A.Goldenberg, E.P., Cuoco, A.A. and Mark, J.: 1998, ‘A role for geometry in general edu-

cation?’, in R. Lehrer and D. Chazan (eds.), Designing Learning Environments forDeveloping Understanding of Geometry and Space, Lawrence Erlbaum Associates,Hillsdale, NJ, pp. 3–44.


Hadas, N. and Hershkowitz, R.: 1998, ‘Proof in geometry as an explanatory and convincingtool’, in Olivier, A. and Newstead, K. (eds.), Proceedings of the 22nd PME Conference,Stellenbosch, South Africa, Vol 3, pp. 25–32.

Hadas, N. and Hershkowitz, R.: 1999, ‘The role of uncertainty in constructing and provingin computerized environments,’ in O. Zaslavsky (ed.), Proceedings of the 23rd PMEConference, Vol 3, Haifa, Israel, pp. 57–64.

Hanna, G.: 1990, ‘Some pedagogical aspects of proof’ Interchange 21(1), 6–13.Hanna, G.: 1995, ‘Challenges to the importance of proof’ For the Learning of Mathematics

15(4), 42–49.Harel, G. and Sowder, L.L.: 1998, ‘Students’ proof schemes: Results from exploratory

studies’, in E. Dubinsky, A.H. Schoenfeld and J.J. Kaput (eds.), Research on CollegiateMathematics Education, American Mathematical Society, Providence, RI, USA, Vol. III,pp. 234–283.

Hersh, R.: 1993, ‘Proving is convincing and explaining’, Educational Studies in Mathem-atics 24(4), 389–399.

Hershkowitz, R., Dreyfus, T., Ben-Zvi, D., Friedlander, A., Hadas, N., Resnick, T., Tabach,M. and Schwarz, B.B.: in press, ‘Mathematics curriculum development for computerizedenvironments: A designer-researcher-teacher-learner-activity’, in L.D. English (ed.),Handbook of International Research in Mathematics Education, Lawrence ErlbaumAssociates, Hillsdale, NJ.

Laborde, C.: 1995, ‘Designing tasks for learning geometry in a computer-based environ-ment, the case of Cabri-géométre’, in L. Burton and B. Jaworski (eds.), Technology inMathematics Teaching – A Bridge Between Teaching and Learning, Chartwell-Bratt,London, pp. 35–68.

Movshovitz Hadar, N.: 1988, ‘Stimulating presentations of theorems followed by respons-ive proofs’, For the Learning of Mathematics 8(2), 12–19; 30.

Senk, S.L.: 1985, ‘How well do students write geometry proofs?’, Mathematics Teacher78(6), 448–456.

Schoenfeld, A.: 1986, ‘On having and using geometric knowledge’, in J. Hiebert (ed.),Conceptual and Procedural Knowledge: The Case of Mathematics, Lawrence ErlbaumAssociates, Hillsdale, NJ, pp. 225–263.

Vinner, S.: 1983, ‘The notion of proof – some aspects of students’ views at the senior highlevel’, in R. Hershkowitz (ed.), Proceedings of the 7th PME Conference, WeizmannInstitute of Science, Rehovot, Israel, pp. 289–294.

Yerushalmy, M. and Chazan, D.: 1990, ‘Overcoming visual obstacles with the aid of theSupposer’, Educational Studies in Mathematics 21, 199–219.

Yerushalmy, M., Chazan, D. and Gordon, M.: 1993, ‘Posing problems: One aspect ofbringing inquiry into classrooms’, in J. Schwartz, M. Yerushalmy and B. Wilson (eds.),The Geometric Supposer, What is it a Case of?, Lawrence Erlbaum Associates, Hillsdale,NJ, pp. 117–142.

Department of Science Teaching,Weizmann Institute of Science,Rehovot,Israel

The Hebrew University,Israel

COLETTE LABORDE

DYNAMIC GEOMETRY ENVIRONMENTS AS A SOURCE OF RICHLEARNING CONTEXTS FOR THE COMPLEX ACTIVITY OF

PROVING

ABSTRACT. Is proof activity in danger with the use of dynamic geometry systems (DGS)?The papers of this special issue report about various teaching sequences based on the use ofsuch DGS and analyse the possible roles of DGS in both the teaching and learning of proof.This paper is a reaction to these four papers. Starting from them, it attempts to develop aglobal discussion about the roles of DGS, by addressing four points: the variety of possiblecontexts for proof in a DGS, the dual nature of proof (cognitive and social) as reflectedin the ‘milieu’ constructed around the use of a DGS, from observing to proving, and theovercoming of the opposition between doing and proving.

KEY WORDS: Deductive geometry, Dynamic geometry, Geometrical Construction, Justi-fication, Learning and teaching proof, Learning environments, Secondary school

Not surprisingly, proof has given rise to many debates among researchersin mathematics education since it is the essence of mathematics and theteaching of proof in mathematics is a key issue which has been investigatedover more than thirty years. Theoretical frameworks have been developedand numerous empirical data have been gathered in experimental settingsinside or outside the mathematics classroom.

Across the world, debates and discussions have risen again with the in-creasing use of dynamic geometry computer environments. As mentionedin all the four papers in this special issue, it has often been claimed that theopportunity offered by such environments to ‘see’ mathematical propertiesso easily might reduce or even kill any need for proof and thus any learningof how to develop a proof.

The four papers decided to investigate this issue and to bring answers.All of them decided to construct answers within a theoretical frameworkand to set up experiments on the basis of their theoretical perspective.Even if their theoretical frameworks have some elements in common, eachpaper focuses on a specific aspect of a proof and the set of the four papersprovides a multifaceted view on the concept of proof in students learning.This issue illustrates perfectly how the process of proving is complex andinvolves many different dimensions and aspects. In the following lines, wetry to describe to what extent the four papers complement each other.


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VARIETY OF CONTEXTS FOR PROOFS IN A DYNAMIC GEOMETRY

ENVIRONMENT

The collection of the four papers illustrates well the variety of possible usesof proof in mathematical activities. While three papers address the learningand teaching of proof for 15–16 year old students at a level of school inwhich proof is part of teaching, the paper by Jones deals with preparingyounger students (12 year old) to deductive proof by making them awareof dependency between properties and enabling them to formulate suchproperties in a mathematical language.

The paper by Jones prepares proof with activities in which studentsmust explain what is observed on the computer screen. The papers byMariotti and by Marrades and Gutierrez report on a long term teachingexperiments in which a social contract is introduced in the class accord-ing to which conjectures or constructions have to be justified: why areconjectures true or why are constructions valid? Hadas, Hershkowitz andSchwarz introduce the need for proof as a way of overcoming contradictionor uncertainty.

The set of papers illustrate the variety of functions and roles of proofwhich have been made explicit in previous theoretical frameworks (Bala-cheff, 1987; de Villiers, 1998; Hanna and Jahnke, 1996). The papers showvery well how proof appears in a diversity of contexts for various reasonsand how the teacher can play on various contexts and situations to motivateproof activities. By means of empirical evidence, the papers refute thecurrent idea of proof being in danger by dynamic geometry environments;the situations and contexts proposed in the papers are actually based on thefeatures of the environment.

The paper of Jones reports on a teaching unit made of three phasesabout the classification of quadrilaterals in which students had reproduce afigure which could not be ‘messed up’ and in some cases satisfying addi-tional conditions. For example, they had to construct a rectangle in such away that by dragging one of its vertices, it could be modified into a square.After constructing the figure the students had to explain why the construc-ted figure was the expected one. Explanation in these tasks prefigures proofin the sense that explaining consists of giving the conditions implying thatthe constructed figure is the expected type of quadrilateral. This task dealswith the idea of implication between properties (or relations) which isnecessary to understand how proof works. The context giving meaningto proof (or rather explanation) is the robustness of a figure under thedrag mode. The explanations provided by the students give mathematical

DYNAMIC GEOMETRY AND THE LEARNING OF PROOF 153

reasons for the fact that a figure remains a specified quadrilateral in thedrag mode.

The paper by Mariotti reports on a long-term teaching experiment inwhich the system of axioms and theorems is constructed by students them-selves as a system of commands introduced in the software which is empty1

at the beginning of the teaching sequence. Proof is the means for justifyingthat the new command will provide the expected outcome. This is achievedby using what is known about previously implemented commands. Theconstruction of the commands of the system is similar to the constructionof theorems, as written by Mariotti: “it is done in parallel with the construc-tion of the theory”. Proof is here a means of being sure that the constructedDG system works as intended. But according to the rules established in theclassroom, every student, confident in the validity of his/her construction,should defend it in front of his/her classroom mates. Proof fulfils thus atwofold role: establishing the validity of a construction for each individualand convincing the other students to accept the construction process.

Although differing from Mariotti’s project, the long-term teaching se-quence of Marrades and Gutierrez presents some common points. It is alsobased on a social organisation in which a solution proposed by a studentmust be accepted by the others. In both cases (Mariotti and Marrades andGutierrez), the discussion is guided by the teacher since proof is not themost immediate social way of conviction among students. The teacheris the warrant of the respect of the discussion and justification rules es-tablished in the class. The students have a notebook of ‘accepted’ resultswhich is updated at any time a new theorem has been proven. This note-book plays the role of the extendable system of commands of the DGSin Mariotti’s classroom organisation. In both papers there is a specificsocial organisation in the classroom for assigning a social role to proofand increasing the need of having recourse to it.

Whereas formulating proof partly emerges in those papers for socialreasons, in the paper of Hadas, Hershkowitz and Schwarz the need forproof is mainly due to cognitive reasons and disequilibria. These authorsvery carefully designed two sequences of tasks in which the order of thetasks led students to develop expectations which turned out to be obviouslywrong when they checked them in the dynamic geometry environment. Itcreated a conflict and an intellectual curiosity to know why this unexpectedproperty is true. In the first activity, the ‘false’ guess was favoured by thefirst task about the sum of the angles of a polygon depending on the numberof its sides. In the second task of the same activity, students were asked toguess the sum of the exterior angles and were thus predisposed to think thatthis sum depends on the number of sides. The second activity is a subtle

154 COLETTE LABORDE

succession of questions about conditions for triangles to be congruent ornot congruent leading students to believe that some constructions werepossible that turned out to be impossible and conversely.

Hadas, Hershkowitz and Schwarz created a climate of uncertainty whichagain compelled in students the need to understand better and not to simplycheck that what they guessed was wrong. It seems at a fist glance that thesame sequence could be written for environments without DGS. Actuallyit would be impossible since the false conjectures came after students wereconvinced of other properties thanks to the DG system. These conjectureswere elaborated in the continuation of valid conjectures. In other casesthe DG environment gave a counter example to an expected result. Thisinterplay of conjectures and checks, of certainty and uncertainty was madepossible by the exploration power and checking facilities offered by theDG environment. Several examples given by the authors show how usingthe drag mode allowed the students to investigate whether it is possible toget non-congruent triangles with a given number of congruent elements.

THE DUAL NATURE OF PROOF AND THE ‘MILIEU’

Although the four papers do not refer to the theoretical notion of ‘milieu’(Brousseau, 1997), it seems worth introducing it to interpret the carefulorganisation of the context of proof production in those papers. To explainthe solving processes carried out by a student in a task, Brousseau proposesto model these processes as resulting from mutual interactions betweentwo systems: the learner and a system offering possibilities of actionsand reactions, a system on which the student can act and which reactsto the actions of the student. This system (called ‘milieu’ by Brousseau)is a theoretical construct, which allows for explaining the strategies of thestudents. As the student strategies are affected by the context, it is clearthat context and milieu are related. However the notion of context refers toall external elements, whilst the notion of milieu accounts for the elementsof material as well as intellectual nature which are not controlled by thestudent and intervene in his/her mathematical behaviours in the task.

Proof is a target knowledge in the papers and the ‘milieu’, offering bothfeedback and action possibilities, includes the dynamic geometry environ-ment in all of them. But a DGS itself without an adequately organisedmilieu would not prompt the need for proof. It is a common feature ofall papers to have constructed a rich milieu with which the student isinteracting during the solving process and the elaboration of a proof. Inall papers, the milieu is developing and at each step in the sequence oftasks is constituted of all the results found at the previous steps.


The teaching unit presented by Jones is constructed according to thisprinciple: at first the students gained understanding of a robust figure inthe drag mode as a figure constructed by means of geometrical relations.It is only because they grasped this idea of robustness that they could befaced with the task of explaining why a constructed quadrilateral is of aspecified type. In the same way, the question “why all parallelograms aretrapeziums ?” (task 7 of phase 3) made sense to them only because inpreceding activities, they could draw trapeziums and transform them intoparallelograms by dragging. Although he does not analyse the successionof tasks in terms of milieu, Jones stresses this idea of progression, byspeaking of ‘progressive mathematisation’ in which “mathematical modelsare developed through the successive positioning of contexts that embodythe underlying structure of the concepts.”

The organisation of the milieu has been achieved in the papers accord-ing to two different ways:

– a cognitive way consisting of a progressive construction of mathem-atical statements by means of tasks and systematically reconsideredand questioned by the following tasks

– a social way consisting of a construction of social rules of acceptanceof results in the classroom.

This is not surprising that the dual nature of the constructed milieu, cog-nitive and social, corresponds to the dual nature of proof, so often stressedby various theoretical perspectives. A proof in mathematics is a specifickind of discourse meant both for validating the truth of a statement andfor convincing the others of the validity of this assertion. In both cases, themilieu is evolving during the sequence of activities and this is the evolutionof the milieu which is a catalyst for proof.

In the paper by Hadas, Hershkowitz and Schwarz it is because the stu-dents knew more about interior angles of a polygon that they could forgeideas of what could be the behaviour of the sum of exterior angles. Theseideas turned out to be false thanks to the feedback from the DGS. Ideas andspeculations did not emerge in a vacuum, they originated from existingknowledge. It is at this point that DGS can play a role in giving evid-ence that a conjecture is not valid. Cognitive conflict and/or surprise (asstressed by Aristotle, wonder is both source and end of knowledge) makethe students eager to understand why. Understanding means grasping themathematical reasons of the observed contradiction. Understanding cannotbe achieved just through visual evidence as understanding requires re-structuring the system of conceptions and ideas. Proof based on theoreticalarguments becomes a means to understand.

156 COLETTE LABORDE

It is because of social rules of discussion for accepting or rejectingstatements that proof becomes a means of convincing the others as faras everybody can check the validity of a proposed reasoning through thesystem of rules. Theory may be the warrant of a democratic debate whilenot being based on authority arguments.

Both kinds of milieu have their limitations:

– students must understand and agree to enter a collective discussion,they must follow the rules and the role of the teacher is of coursecritical with regard to this aspect

– a cognitive conflict may not arise or if it arises, it may not lead toovercoming it. In the experiment of Hadas, Hershkowitz and Schwarz,around 20% of the answers in each of the activities did not mentionany explanation but just stated affirmations.

It is also not surprising that in both cases, the organisation of the mi-lieu is based on memory of the individual student and of the classroom(Brousseau and Centeno, 1991). Memory of what is already known, memoryof what is already accepted. Proof in mathematics is by essence basedon memory. Axioms and theorems constitute a collective memory of themathematician’s community on which they rely to go further and to pro-duce proof of new statements. The deductive reasoning is based on memory.Memory is even reified in two teaching sequences under the form of thestate of the DGS commands (Mariotti) and of a note book (Marrades andGutierrez).

As soon as memory is part of the functioning of the milieu, time be-comes a critical variable. Memory is not instantly built, memory requiresnot only accumulation of data but also sorting, eliminating and structuringof data. Memory is a process over time. The papers do not develop extens-ive comments on this dimension but actually this latter plays a decisiverole in the evolution of students. The two papers involving collective dis-cussions are based on long-term teaching sequences. Installing social rulesrequires obviously time. The system of tasks in all papers progressed ata pace related to the cognitive evolution of students. “The complexity ofthe theoretical system increased at a rate which the pupils were able tomanage” (Mariotti). Hadas, Hershkowitz and Schwarz managed differentpaces in the sequence of the tasks of the congruence activity depending onthe answers of the students.


CONCEPTUAL BREAK VERSUS CONTINUUM IN THE MOVE FROM

OBSERVING TO PROVING

It has been often said in the past that deductive geometry differs deeplyfrom geometry of observations: in the former every assertion is either agiven or must be deduced from the givens, in the latter, what you see inthe figure can be taken as granted. We would certainly claim that thereis a deep change of status of the objects in the move from the geometryof visual evidence to the geometry of objects and relations involved in adeductive system as expressed in numerous research papers and confirmedby empirical evidence. But the existence of this conceptual cut does notimply that the former knowledge of students is not useful when faced withnew task of proving. Nor does this imply that the solving processes of aproof problem are purely deductive, as we will come back to this pointbelow.

The paper by Jones is very appropriate to illustrate how proof can beprepared in teaching with activities aimed at developing students’ aware-ness of dependency between properties. If properties of figures are notconceived as dependent, a deductive reasoning has no meaning. The ques-tion of the validity of a property conditional on the validity of other prop-erties would not arise in a world of unrelated properties. The sense ofnecessity links between properties must be developed to give a meaningto that question. Constructing robust figures under drag mode may revealthese necessary links. As soon as one constructs a parallelogram with fourequal sides, one can observe that its diagonals remain perpendicular in thedrag mode. However it may happen that this observation does not lead toconjecture a dependency between rhombus and perpendicular diagonals.It is interesting to note that in phase 2 devoted to construction tasks ofrobust figures under drag, students did not really formulate the depend-ency between properties: “It is a square because the sides are equal andthe diagonals intersect. The diagonals are at right angles (90◦)” (pair A).But in phase 3 where they had to modify a rectangle into a square and toexplain why all squares are rectangle, they extracted the relevant additionalproperty, which transforms a rectangle into a square. “A rectangle becomesa square when the diagonals become right angles where they meet” (samepair A).

Jones stresses this change in the formulations of students from purelydescriptive relying on perception to more precise explanations, at firstsituated in the dynamic geometry environment and then related to themathematical context. We see phase 3 as critical in this move, even ifthe formulations of students in this phase were referring to movement or

158 COLETTE LABORDE

drag. In the movement (change of space over time), the conjunction oftwo arising phenomena at a given time revealed to the students the linkbetween the two phenomena for a rectangle: to become a square and tohave perpendicular diagonals. In a robust construction under drag, the factthat all valid properties remain satisfied, is certainly a good indication thatthere is a dependency relation for experts or students aware of necessarylinks. It is not the case for novice students oscillating between a world ofunrelated properties and a more structured world in construction. Change ismore evident to be perceived by novices than permanence. This is why thesimultaneity in the change of appearance is critical for novices because itis a strong external sign of a link between the two objects changing exactlyat the same instant.

The continuity in the overcome of the break between empirical andformal ways of justification is present in all four papers, be it in the or-ganisation of the tasks as in Mariotti or Hadas, Hershkowitz and Schwarzor in the solving processes of the students as in Marrades and Guttierez.These latter advocate in favour of the idea of a long and slow transitionfrom empirical to formal justifications as reflected in their fine analysisof students solving problems in a DGS environment. They show how thedeductive phase does not appear at the beginning of the solving process butafter several empirical approaches and when it appears, how it is related tothese empirical approaches.

A question arises from this convergence among the four papers. Wouldthe continuity from empirical to more deductive not particularly be sup-ported by DGS in that DGS offer a break with paper and pencil geometry?DGS contain within them the seeds for a geometry of relations as opposedto the paper and pencil geometry of unrelated facts. The break would bealready entailed in the use of the DGS. This could explain why studentsdo not enter immediately the new contract of construction of figures ina DGS, they must learn it. Instead of being faced with this the break atthe level of formulations, students are faced earlier at the level of actions.We hypothesise that overcoming the break in action is easier for them for atleast two reasons: from a cognitive point of view action very often precedesformulation, and feedback to actions is more easily recognised.

The continuity from empirical to deductive is reflected in the four pa-pers by the interrelations between doing and proving which are visible inthe behaviours of the students.


OVERCOMING THE OPPOSITION BETWEEN DOING AND PROVING

This interaction between student and the DG environment which has beenorganised in all the papers of this issue illuminates how proof is not sep-arated of action. Hadas, Hershkowitz and Schwarz show very well how“some explanations had to be expressed by doing” and how “action pushedstudents towards deductive explanations”. The fundamental principle un-derlying the teaching sequence in the paper by Mariotti relies on this dia-lectical link between action and proof, or more precisely between con-struction and proof. Mariotti explains in the example of students G and Chow the command ‘carry an angle’ is both a construction command andan internal tool related to a theoretical control. The action of construct-ing a congruent angle in all possible ways in the triangle activity (Hadas,Hershkowitz and Schwarz) led the students to understand that there arenon congruent angles with one congruent side and two congruent angles.Hoyles (1998) in a similar activity of investigating the minimal informationto construct a triangle that is congruent to the one given, concluded aboutthe matching between proof and construction.

Marrades and Guttierez develop a fine analysis of the interaction betweenthe ascending phase in the solving process “characterised by an empiricalactivity” and a descending phase “where the solver tries to build a de-ductive justification”. In several examples they show the intertwining ofconstruction and proof. Students H and C closed the shape built by twoparallel lines and a segment (Figure 1) by constructing a parallel line tosegment AC and obtained thus a rhombus (Figure 2). This probably wasjust meant for satisfying a visual demand (according to the Gestalt psy-chology). But the shape rhombus allowed the students to recognise a lineof symmetry (AK). This was the starting point of a construction leading totake into account one of the hypotheses of the problem statement (AB=AC)and to use congruence of triangles to deduce the expected property (BC isangle bisector of angle ACD).

Figure 1.

160 COLETTE LABORDE

Figure 2.

Another example (the example of T and P), carefully reported in Mar-rades and Guttierez illuminates very well how a deductive justificationtakes its roots in a succession of trials just using drag mode and perceptivecontrols and of robust constructions meeting the conditions of the prob-lem. The authors conclude to the dual nature of the justification: empiricaland intellectual (“empirical justification by intellectual generic example”).They write: “The successive constructions added conceptual elements thathelped the students to recognise and connect the different mathematicalproperties necessary to obtain the correct figure and, then, to justify itscorrectness”. The new categorisation of proofs proposed by the authorsreflects this complexity of the proving process made of various kinds ofapproaches.

THE ROLE OF A DYNAMIC GEOMETRY ENVIRONMENT

Without doubt the dynamic geometry environment fostered this interactionbetween construction and proof, between doing on the computer and jus-tifying by means of theoretical arguments as claimed by Hoyles (op.cit.):“Some commentators may question whether the presence of the computerwas necessary [. . .] it was to make construction methods explicit, to al-low reflection on properties, to check things out and obtain immediatefeedback but most crucially to foster [. . .] an experimental atmospherethat the teacher could exploit to introduce formal proofs in ways whichmatched rather than supplemented student constructions”. We would fur-ther argue that dynamic geometry environment leads to analyse differentlythe processes involved in a proving activity as mentioned in Hadas, Her-shkowitz and Schwarz and illustrated by the new categorisation of proofsproposed by Marrades and Guttierez. Following Jones and Mariotti, wecould also claim in Vygotskian terms that DG environments afford possib-ilities of access to theoretical justifications through the semiotic mediation


organised by the teacher around construction tools of dynamic geometryenvironments.

NOTES

1. Cabri allows the user to configure the available tools and menus by adding any newconstruction or suppressing any tool.

REFERENCES

Balacheff, N.: 1987, ‘Processus de preuve et situations de validation’, Educational Studiesin Mathematics 18, 147–176.

Brousseau, G.: 1997, Theory of Didactical Situations in Mathematics, Didactique desMathématiques 1970–1990, Kluwer Academic Publishers, Dordrecht.

Brousseau, G. and Centeno J.: 1991, ‘Rôle de la mémoire didactique de l’enseignant’,Recherches en didactique des mathématiques 11(2–3), 167–210.

Hanna, G. & Jahnke, N.: 1996, ‘Proof and proving’, in A. Bishop (ed.), InternationalHandbook of Mathematics Education, Kluwer Academic Publishers, Dordrecht, pp.877–908.

Hoyles, C.: 1998, ‘A culture of proving in school mathematics’, in D. Tinsley and D.Johnson (eds.), Information and Communication Technologies in School Mathematics,Chapman and Hall, London, pp. 169–181.

de Villiers, M.: 1998, ‘An alternative approach to proof in dynamic geometry’, in R. Lehrerand D. Chazan (eds.), Designing Learning Environments for Developing Understandingof Geometry and Space, Lawrence Erlbaum, Mahwah, USA, pp. 369–393.

University Joseph Fourier & University Institute for Teacher Education,Grenoble,France

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Educational Studies in Mathematicsmath.fau.edu/yiu/PSRM2015/yiu/New Folder (4)/Downloaded Papers... · Proof is an important part of mathematics itself, of course, and so we must