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[1]. 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[2] draws the cube in Fig.1 and solves the problem correctly.

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."

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.

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.

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[3]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[4].

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: