DEFINITION AS A TEACHING OBJECT: A PRELIMINARY STUDY



Revisiting Guided Reinvention: icons, indexes, and symbols in the construction of mathematical objects

DRAFT VERSION

Fulvia Furinghetti* & Domingo Paola**

*Dipartimento di Matematica dell’Università di Genova, Italy

**Liceo Scientifico ‘A. Issel’, Finale Ligure (Sv), Italy

abstract. In this paper we discuss various types of signs (mainly icons and geometrical figures) used by students in constructing mathematical objects and in proving.

introduction

IN THIS PAPER WE CONSIDER THE CONSTRUCTION OF MATHEMATICAL OBJECTS IN THE CLASSROOM. THE TERM 'MATHEMATICAL OBJECTS' HAS TO BE INTENDED IN A BROAD SENSE INCLUDING DEFINITIONS AND PROOF. ABOUT PROOF OUR APPROACH TAKES INTO ACCOUNT THE FOLLOWING ELEMENTS:

• focus on the debate about the construction of theorems and proofs, with the distinction between the problems of conforming to the standards of exposition and rigor and those of construction, validation, and acceptation of a statement

• specification of the rules for stating a theorem and proving it

• reflection on the environments which seem to foster the production of hypotheses and their formulation according to logical connection

• possibility of singling out cognitive continuity between the processes of producing and exploring the statement of a theorem and the construction of its proof, with particular attention to the related theories of and the leaps inside a theory and among different theories

• the role of the social dimension of the learning as for knowledge on theorems and proof, with particular reference to mathematical discussion in classroom and the modes of using various mediators (history, technology,…).

The current international debate among mathematics educators around proof has shown that such an introduction is a hard task for many different reasons. One of the difficulties consists of the fact that, as Moore (1994) points out in the case of the United States, “the transition to proof is abrupt” and this “abrupt transition to proof is a source of difficulty for many students, even for those who have done superior work with ease in their lower-level mathematics courses” (p.249). The same problem is encountered in many different countries (including ours) and at different school levels. Another factor which makes the transition to proof so difficult is the typical classroom environment, described by Lampert (1990, p.32] as follows:

In the classroom, the teacher and the textbook are the authority, and mathematics is not a subject to be created or explored. In school the truth is given in the teacher’s explanations and the answer book; there is no zig-zag between conjectures and arguments for their validity, and one could hardly imagine hearing the words maybe or perhaps in a lesson.

The classroom style we advocate to promote the transition to proof is conveniently described by a metaphor of Pollak, quoted in (Lampert, pp.41-42): “to move around in mathematical territory in a flexible manner”, that is to say to do a kind of “cross-country” mathematics, instead of “walking on a path that is carefully laid out through the woods”. Sharing this teaching perspective might make the way of doing mathematics in the classroom closer to the way mathematicians do. In this concern Schoenfeld, (1992), going along the work of Polya and Fawcett, discusses many examples of this way of engineering the class according to these ideas. For example, it has to be avoided to begin the assignments with the expression "prove that…". Rather it is recommended to present the task in a way which fosters the classroom discussion orchestrated by the teacher. Moreover, also cultural and social processes are integral to mathematical activity: the way in which students are required to participate in the classroom life may affect students’ transition to proof.

We think that only through experimenting personally the construction of parts of a theory (under the guidance of the teacher and in situations carefully projected) students may give up, when necessary, the perceptual level and appreciate the meaning of theories. To make students to construct parts of a theory means to allow them to experience the construction of mathematical knowledge at different levels: exploring within particular cases, observing regularities, producing conjectures, validating them inside theories (which may be already constructed or in progress). In developing this approach we are concerned with the transition from elementary to advanced mathematical thinking. Gray et al. (1999) have pointed out that the “didactical reversal – constructing a mental object from ‘known’ properties, instead of constructing properties from ‘known’ objects causes new kinds of cognitive difficulty.” (p.117)

Nunokawa (1996) has discussed the application of Lakatos’s ideas to mathematical problem solving. In our work on proof in classroom we are taking a similar approach. We see students as immersed in a situation close to that termed by Lakatos (1976) pre-Euclidean, that is to say a situation in which the theoretical frame is not well defined so that one has to look for the ‘convenient’ axioms that allow constructing the theory. The didactical suggestion implicit in Lakatos’s words is that it is advisable to recover the spirit of Greek geometers. When they made proofs they were not inside a theory in which axioms were explicitly declared. Initially antique geometry developed in an empirical way, through a naïve phase of trials and errors: it started from a body of conjectures, after there were mental experiments of control and proving experiments (mainly analysis) without any sure axiomatic system. According to the comment of Szabó, this is the original concept of proof held by Greeks, called deiknimi. The deiknimi may be developed in two ways, which correspond to analysis and synthesis. Only after a lot of successful processes of analysis and synthesis, after a lot of proofs (in the sense of proofs and refutations) some lemmas, which were repeatedly used, became stronger than others, while their alternative remained sterile. These lemmas became the core of Euclid's program ('axiomatic system'). Since then when a geometric conjecture was suggested, the problem was to establish if it followed from Euclidean axioms and postulates, and not only if it was 'true'. Szabó claims that the most interesting analysis of Greek geometry were pre-Euclidean and their role was to generate the axiomatic system of Euclid. The most Euclidean geometry existed before postulates, axioms, definitions, and common notions of Euclid.

We do not want to go too deeply in the historical discussion on the genesis of Euclidean geometry, which encompasses different positions. What interests us is the didactical insight offered by this historical interpretation. It suggests the question: what is the meaning of reproducing in school Euclidean geometry (or, even worst, Hilbert geometry) already set in an axiomatic system if students have not grasped the gradual generation of an axiomatic system? In Lakatos's position we find a possibility of cognitive continuity versus the present discontinuity existing in the style of proving in classroom. In the same time the possible tools to act according to these lines are suggested: socialization, sharing ideas, discussion. The classroom discussion orchestrated by the teacher have to lead gradually students from argumentation used to convince that their conjecture is true to a proof which explains why it is true This may be done through a (re)construction of a system of axioms. At a certain stage this (re)construction is mainly consisting in making explicit their own knowledge, beliefs, prejudices through negotiation and discussion. The mediator are, of course, the teacher as well as historical sources, microworlds.

We feel that Lakatos's ideas better apply to teaching than to mathematical research. In classroom we have a position similar to that of the early geometers, a situation in which knowledge begin to form. We have not structured science or knowledge and students have to construct it gradually. This is our way of revisiting guided reinvention of Freundenthal: to put students in the situation of the pioneer mathematicians and to create a context suitable to construct the mathematical object through socialization, discussion, sharing of ideas.

The situation in which we involve our students allows to scrutinize the process through which they build mathematical objects and approach proof. In this paper we focus on the role of signs in this construction. We know that Peirce distinguishes among three kinds of signs: - icon, i.e. something which designates an object on the ground of its similarity to it; - index, i.e. something which designates an object pointing to it in some way; - symbol, which designates an object on the ground of some convention. We will see that students use all these kind of signs, in different manner and in different situations. In particular we will pay attention to the status of diagrams in students' learning. In this concern we find very illuminating the paper of Netz (1998) on the status of diagrams in Greek mathematics. Netz's results may be a useful reference for our work. According to this author in Greek mathematics "Determination of objects is done through the diagram. […] The diagram is not just a pedagogic aid, it is necessary, logical component" (p.34) In this perspective the diagram is an essential part of the text. Our research questions are: "Do diagrams play an analogous role in present students' work?", or, to put it in a more general way, "Which role signs play in students work?".

icons and indexes

THE FIRST EXAMPLE WE DISCUSS SHOWS THAT SIMPLE ICONIC REPRESENTATIONS MAY BE ESSENTIAL COMPONENTS OF THE REASONING. IN AN EXPERIMENT THE FOLLOWING PROBLEM HAS BEEN GIVEN TO 37 STUDENTS AGED ABOUT 15 (ITALIAN SECONDARY SCHOOL) AND 5 UNDERGRADUATE STUDENTS

Given a cube made up by little cubes, take away a full column of little cubes. The number of the remaining little cubes is divisible by six. Try to explain why this happens.

The work was carried out in group of three students, with the teacher and an exterior observer. The analysis is based on protocols, and fieldnotes of the observer[i]. All students but two have drawn a figure, representing a cube in the plan of the sheet. This was a typical use of icon as sign. The function of the icon was merely that of translating the text in a language more telling. For most students the figure presented in the icon mode was just the starting point for exploring one or a few numerical examples; afterwards the reasoning shifted from the figural context into the algebraic context.

In some cases the drawing was essential to the development of the reasoning. It pushed the students towards powerful forms of thinking such as transformational reasoning. In this concern we report on a group that followed as an example this pattern. A member of the group, Andrea[ii] draws the cube in Fig.1 and solves the problem correctly.

[pic]

Fig.1

Nevertheless he is not satisfied by the pure 'formal' solution. He writes (Fig.2) "I would have liked to find a proof only with numbers."

[pic]

Fig.2

At this point he looks at the figures and begins to make gestures by hands until he finds the solution in a new way, based on the decomposition and composition of the original cube until a parallelepiped is obtained. In this case the function of gestures of the hands is a means to enhance transformational reasoning. In the very word of the student we see that the drawing is a carrier of meaning, while the algebraic symbol hides the meaning. The icon may be considered a translation of the solution of the problem. Andrea reflects on it and finds its logical component, i.e. realizes the process of analysis. In our case the gestures have been the means to develop it. Simone, another member of the group, draws in the protocol the process described with gestures by Andrea, see Fig.3.

The process of reasoning encompasses icons and gestures, but the mode is symbolic, even if based on perceptual aspects. The student, indeed, considers and manipulates the pieces of cubes as representatives of the algebraic symbols x, x-1, x+1.The process conceived by Andrea has extraordinary resemblance with the process of "cut and paste" realized by Al-Khwarizmi (1831; 1838) for solving second degree equations.

[pic]

Fig.3.

signs in geometry

IN THE GEOMETRIC CONTEXT DIAGRAMS ARE THE MOST USED SIGNS. THEIR STATUS VARIES ACCORDING TO DIFFERENT SITUATIONS. FOR EXAMPLE, THE SKETCHES WE MAKE AFTER READING A GEOMETRIC STATEMENT ARE ICONS. THE DRAWINGS FOUND IN THE TEXTBOOKS ARE PERCEIVED BY STUDENTS AS ICONS. WE MEAN THAT STUDENTS TAKE FOR GRANTED THAT, FOR EXAMPLE, THE DIAGRAM WHICH IS SAID TO REPRESENT AN ISOSCELES TRIANGLE IS AN ISOSCELES TRIANGLE BECAUSE IT LOOKS AS THAT. STUDENTS DO NOT REFER TO A PARTICULAR CONSTRUCTION WHICH MAY HAVE GENERATED IT. THIS KIND OF ICON` PUTS THE MATHEMATICAL PROPERTIES OF THE OBJECT IN THE SHADE AND FOCUSES ON THE PERCEPTUAL ASPECTS. RECALLING THE PROPERTIES OF THE FIGURE AND TO USE THEM IS AN OPERATION THAT STUDENTS PERFORM ON THE GROUND OF VERBALIZATION AND RATHER INDEPENDENTLY FROM THE ICON THEY HAVE SKETCHED. THE INFLUENCE OF PERCEPTUAL FACTORS IN DEALING WITH SUCH A KIND OF FIGURES IS STRONG AND CARRIES SOME DISADVANTAGES WELL KNOWN IN LITERATURE. SCHOENFELD (1992) REPORTS THAT STUDENTS WHO KNOW THE PROPERTIES OF TANGENT LINES IN A GIVEN POINT OF A CIRCLE WITH GIVEN CENTER, WHEN DEALING WITH FIG.4 TEND (30%) TO PUT THE CENTER OF THE CIRCLE IN THE MIDDLE POINT OF THE SEGMENT JOINING THE TWO TANGENT POINTS. IN THIS CASES THE ICON HIDES THE PROCESS OF CONSTRUCTION AND THUS PREVENTS THE ACTIVATION OF THE PROCESS OF ANALYSIS (IN THE GREEK SENSE) OF THE FIGURE.

[pic]

Fig.4

In old books of geometry addressed to artists and architects it was common to train students to construct geometric diagrams; the whole construction was kept in the diagram reported in the book, see (Leclerc, 1761). In this case the figure is the end of the process, the synthetic thinking is activated, but the figure is scarcely suitable to be used as a tool for solving a further problem, since all the attention is concentrated on the very construction. For its genesis this kind of diagram has a static character, since a new diagram implies to repeat all process of construction.

Dynamic geometric software[iii]introduces a new kind of figures, which are the result of geometrical constructions without keeping 'apparent traces' of these constructions so that they function as icons produced by sketches. On the other hand the dragging mode allows to recall in any moment the properties on which these figures rely and prevents to trust only on perceptual features. When Cabri is used to draw an isosceles triangle the figure looks as an isosceles triangle and is an isosceles triangle. It is well known that dynamicity is the main character of figures obtained through Cabri.

The different status of the geometric diagrams obtained with Cabri changes students' behavior in constructing mathematical objects and in proving. The few examples we report in the following illustrate some aspects.

In (Furinghetti & Paola, 2002) we report an experiment which involved 21 Italian 10-grade students with previous experience of using Cabri. They worked in pairs (one PC per pair), except one group of three students. The teacher and a researcher acting as an observer (he was one of the authors) were present. The observer was not passive, but talked with students and addressed their activity to Cabri in order to obtain more information about the interaction student-Cabri. The task given to students was to produce a classification of quadrilaterals.

To perform the task all students used the computer (even if it was not compulsory). All groups began with the construction of a square. The most common sequence was to draw a circle and a square inscribed or circumscribed. Six (on the ten groups) made drawings which were icons, but one student asked himself “How may I decide that this is really a parallelogram?” This is a question that would not have arisen without Cabri.

The students of another group based their construction on symmetries and produced definitions which were construction-oriented. This kind of definitions is better conceived with Cabri. A student constructed rhombuses starting from the properties of diagonals. And, when he attempted to inscribe a given rhombus in the circle that he had drawn at the beginning, he discovered that this is possible on particular conditions. Again a statement to be proved was spontaneously generated by the activity with Cabri. As we have reported before, the environment fostered the use (even limited) of symmetries, which are taught, but are rarely used by students when working with paper and pencil.

In classifying quadrilaterals all students started from the square. A paradigmatic explanation of this fact found in a protocol is:

I started from the square because it is the quadrilateral with more properties and because I can imagine it more easily than the other quadrilaterals.

Other students said that “The square is the easiest quadrilateral”. The generic is more difficult to be conceived than the particular. Figures with regularities are conceived as specified and thus more easily perceived. This fact is even more fostered by dynamic geometric software which allows to construct easily regular geometric figures.

The dynamic geometric environment, indeed, has oriented to a different criterion of classification, which we may term “by default”. It is a kind of reverse hierarchy: one starts from the more specified figure (the square) having the greatest number of properties and goes on by dropping some properties, see Fig.5[iv].

[pic]

FIG.5

WE NOTE THAT THIS PATH WAS FOLLOWED IN EUCLID'S ELEMENTS TOO (BOOK I, 22):

Of quadrilateral figures, a square is that which is both equilateral and right-angled; an oblong that which is right-angled but not equilateral; a rhombus that which is equilateral but not right-angled; and a rhomboid that which has its opposite sides and angles equal to one another but is neither equilateral nor right-angled. And let quadrilaterals other than these be called trapezia

A similar situation in which the use of Cabri orient towards a particular behavior (again the circle as a primitive figure) is described in (Furinghetti et al., 2001). The following problem, taken from (Arsac et al., p.48), was given to the students of a classroom aged 16-17 years:

You are given a right-angled triangle ABC, AB being the hypotenuse. Take a point P on AB. Draw the parallel lines to AC and BC through P. Name H and K the points of intersection with AC and BC respectively. For which position of P does the line HK have minimum length?

The students were working with paper and pencil. We consider the case of a group of three students who succeeded in solving the problem. They were videotaped. At the beginning they consider particular cases. One of the students uses the ruler and moves it in order to show how the length of the segments varies. This may be considered an actual exploratory behavior, which starts from concrete/experimental situations. Another student considers a pencil of circles centered in C and going through P. The circle whose radius is minimum is the circle tangent to the segment AB; therefore HK has minimum length when P is the point of contact (point of tangency) between AB and the circle centered in C. This idea came from applying Pythagoras’s theorem to the triangle PHK, in a system of co-ordinates centered in C, which produced the formula

[pic]

Figure 6 shows some drawings taken from the papers this group handed in.

This group shows a dynamic approach to the problem, which could be interpreted as transformational reasoning. We remind the reader that these students have been using Cabri before and it seems that they are able to transfer the skills and the different way of looking at problems acquired working in Cabri to other situations. For example, a typical feature, which is borrowed from the work in Cabri is the use of circles, which are not so much exploited when working with paper and pencil. The students also transfer from Cabri the dynamic way of looking at the problem; this way provides different models of a given situation, which may foster generalization.

[pic]

Fig.6

In the previous experiment the figure was an optional help to construct mathematical object. In other experiments such as that reported in the following, see also (Furinghetti & Paola, 2003), the figure is essential to reach the solution. The experiment was carried out in a class of a Scientific Lyceum (an Italian high school with a scientific orientation), at the beginning of the school year. This experiment is an example of what may happen in classroom when our approach is proposed. The class begins with a work of exploration and observation, which leads to produce conjectures. The validation of these conjectures is performed through the dragging text with Cabri. The way of reasoning is similar to that employed in empirical sciences, e.g. induction, abduction, analogy. In this context the role of proof is to explain why the produced conjectures hold within a theory (in our case Euclidean geometry).

The students (17 years old) worked in small groups (2 or 3 persons per group, 8 groups) with one computer per group. They were acquainted with exploration of open problems and had worked in group quite regularly before the experiment. Also, they mastered Cabri. The time allowed for the experiment has been one hour and a half. In the following we describe the main phases of the work of the pair composed by Alex and Luca. The report is based on fieldnotes taken by the teacher (who acted as an observer) and on the students’ protocols. The problem assigned was:

You are given a quadrilateral ABCD. Consider the bisectors of the four interior angles: be H the intersection point of the bisectors in ∠A and in ∠B, K the intersection point of the bisectors in ∠B and in ∠C, L the intersection point of the bisectors in ∠C and in ∠D, M the intersection point of the bisectors in ∠D and in ∠A.

Investigate how KHLM changes in relation to ABCD? Prove your conjectures.

We focus on the work of the group of Alex and Luca, who showed an interesting way of reasoning. The students use Cabri: draw quickly and accurately the quadrilateral ABCD and afterwards the interior quadrilateral HKLM. Afterwards they drag the vertexes A, B, C, D at random; in doing this they arrive at a figure in which the interior quadrilateral HKLM becomes a point.

At this moment the mode of dragging changes significantly. Alex and Luca decide to focus on the internal quadrilateral HKLM. Of course, they can only act on the vertexes A, B, C, D, but they choose a particular configuration of KHLM (a point, a square, a rectangle, a rhombus, a parallelogram, a trapezium) and afterwards drag the vertexes A, B, C, D so that the quadrilateral KHLM keeps the particular configuration they have chosen.

These students are the only ones in the classroom using this mode of dragging based on the internal quadrilateral. This allows them to see very soon that not only squares and rhombuses generate internal quadrilaterals that are points (In the case of squares and rhombuses the bisectors of opposite angles are coincident). The function of Cabri in this phase is to support transformational reasoning, see (Simon, 1996).

We stress that this phase marks a leap in the exploration. The mode of working goes back from the final result (a particular configuration of the internal quadrilateral) to the premises (the given quadrilateral and the bisectors of angles). This recalls the method of analysis. This method is already present in Pappus's work. There are deep discussions on the real meaning of the Greek word 'analysis', see (Hintikka & Remes, 1992), but for our purposes it is enough the operative description given by Polya (1945; 1966) of analysis as a backward process from what is sought to the premises. We stress the fact that the figure made with Cabri pushed the student towards analysis.

Our students show to be aware of the reverse path followed in their reasoning (“We have looked for a relation among the figures obtained by means of the same internal figure [italic added]”). Since they are convinced of the validity of their conjecture they are motivated to answer the question “Why the conjecture is valid?” In their protocols Alex and Luca use the words “theorem” and “formalized”, which evidence that they have definitely put themselves inside the theoretical framework of Euclidean geometry. They seem to perceive the function of proof as a process suitable to explain why a given conjecture is true. There is one sentence in their writing that shows the interlacement between the exploration (“In this search [made with Cabri]”) and the theory (“we have discovered a theorem”).

After this exploration Alex and Luca are ready to prove the conjecture produced. They abandon Cabri and use paper and pencil, also for drawing the figure on which their proof is based. The proof is not complete and precise, but it could become acceptable with few amendments. In this phase it is clear that the mode of communication is changed. The focus has shifted from the facts observed in the screen to their justification in Euclidean geometry. The transition from the computer to paper and pencil marks the transition to the synthetic mode of proving. The construction of figures made with paper and pencil plays the role of pivot between analysis and synthesis. At the end the students are able to state that the internal quadrilateral becomes a point when the exterior quadrilateral may be circumscribed to a circle. The final statement they write is:

We know the theorem stating that a quadrilateral may circumscribe a circle when the sums of its opposite sides are equal (AB+CD=AD+BC). Hence we may say that H, K, L, M are coincident when AB+CD =AD+BC.

Thus the final statement is expressed in a form (“…when AB+CD=AD+BC”) that hides the steps through which students reached the statement. Definitely the students are in the synthetic mode of reasoning inside the Euclidean theory.

some preliminary conclusions and hints for developments

NETZ (1998) STATES THE FOLLOWING ASSERTION IN DISCUSSING THE USE AND THE MEANING OF THE GREEK DIAGRAMS

Assertion 1: Determination of objects is done through the diagram. (p.34)

Assertion 2: The writing down is preceded by an oral dress rehearsal. That is to say that the process was composed of three stages: (i) drawing a diagram; (ii) a dress rehearsal in front of the diagram, in which the diagram is dressed, i.e. letters are inserted, (iii) a full production, writing down the proof.

Assertion 3: Letters are indices, not symbols. (p.38)

Assertion 4: The diagram is the metonym of the proposition (p.38)

Assertion 5: Mathematics requires an inter-subjectively given object. This is supplied in modern mathematics by conceptual systems and in Greek mathematics by the diagram. (p.38)

Assertion 6: The verbal could not be the fixed object in Greek mathematics because of the more oral approach taken in Greek mathematics. (p.39)

We feel that the preceding assertions apply quite well to the diagrams used by the present students in constructing mathematical objects. They may be taken as a means to explore how students approach the formal thinking when using diagrams obtained through geometric constructions. We see that, in this case, the reinvention is guided, among other things, by the use of Cabri.

references.

AL-KHWARIZMI: 1831, THE ALGEBRA OF MOHAMMED BEN MUSA, F. ROSEN (TRANSLATOR AND EDITOR). LONDON: ORIENTAL TRANSLATION FUND.

Al-Khwarizmi: 1838, Liber Maumeti filii Moysi alchoarismi de algebra et almuchabala incipit (Latin translation), in G. Libri, Histoire des Sciences Mathématiques en Italie (253-297). Paris: Renouard.

Arsac, G., Germain, G. & Mante, M.: 1988, Problème ouvert et situation-problème, IREM, Villeurbanne.

Cartiglia, M.: 2003, Il problem solving come sfida culturale a fare matematica, Tesi di laurea A.A. 2002-2003 Dipartimento di Matematica dell'Uniersità di Genova (Supervisors: F. Furinghettti & D. Paola).

Freudenthal, H.: 1973, Mathematics as an educational task, D. Reidel, Dordrecht.

Furinghetti, F., Olivero, F. & Paola, D.: 2001, ‘Students approaching proof through conjectures: snapshots in a classroom’, International Journal of Mathematical Education in Science and Technology, v.32, 319-335.

Furinghetti, F. & Paola, D.: 2002, ‘Defining within a dynamic geometry environment: notes from the classroom’, in A.D. Cockburn & E. Nardi (editors), Proceedings of PME 26 (Norwich), v.2, 392-399.

Furinghetti, F. & Paola, D.: 2003, ‘to produce conjectures and to prove them within a dynamic geometry environment: a case study’, in Proceedings of PME 27 (Honolulu).

Gray, E., Pinto, M., Pitta, D. & Tall, D.: 1999, ‘Knowledge construction and diverging thinking in elementary and advanced mathematics’, Educational studies in mathematics, v.38, 111-133.

Heath, T.: 1956, Euclid's Elements, v.I, Dover Publications, Inc., New York.

Hintikka, J. & Remes, U.: 1974, The Method of Analysis, Reidel, Dordrecht/Boston.

Lakatos, I.: 1976, Proofs and Refutations: The Logic of Mathematical Discovery, C.U.P., Cambridge.

Lampert, M.: 1990, ‘When the problem is not the question and the solution is not the answer: mathematical knowing and teaching’, American educational research journal, v.27, n.1, 29-63.

Leclerc, S.: 1761, Pratica di geometria in carta e campo per istruzione della nobile gioventù, edizione seconda, Monaldini, Roma.

Moore, R., 1994, 'Making the transition to formal proof', Educational Studies in Mathematics, v.27, 249-266.

Netz, R.: 1998, 'Greek mathematical diagrams: their use and their meaning', For the Learning of Mathematics, v.18, 33-39.

Nunokawa, K.: 1996, ‘Applying Lakatos’ theory to the theory of mathematical problem solving’, Educational Studies in Mathematics, v.31, 269-293.

Polya, G.: 1945, How to solve it, a new aspect of mathematical method, Princeton U.P., Princeton, NJ.

Polya, G.: 1966, Mathematical discovery, John Wiley & Sons, New York.

Proclus: 1978, Commento al Primo Libro degli Elementi di Euclide, Giardini Editori e Stampatori, Pisa.

Schoenfeld, A. H.: 1992, ‘Learning to think mathematically: Problem solving, metacognition and sense making in mathematics’, in A.D. Grows (editor), Handbook of research on mathematics learning and teaching, 334-370.

Simon, M.A.: 1996, ‘Beyond inductive and deductive reasoning: the search for a sense of knowing’, Educational Studies in Mathematics, v.30, 197-210.

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[i] For a full account on this experiment see (Cartiglia, 2003)

[ii] In Italian Andrea is a male name.

[iii] In the following we refer to experiments carried out with the dynamic geometric software Cabri-géomètre. Most of our considerations apply also to other software of this kind.

[iv] Quadrilateri = quadrilaterals, Quadrato = square, Rombo = rhombus, Rett. = rectangle, Parall. - parallelogram, Trap. = trapezium.

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