Boero P., Garuti R., Lemut E., Mariotti M. A. (1996)
Challenging the traditional school approach to theorems: a hypothesis about the cognitive unity of theorems.

PME XX. Valencia, Spain.

Abstract
The purpose of this report is that of highlighting the possibility that in an adequate educational context the majority of grade VIII students successfully implement a process of theorem (conjecture and proof) production, characterised by a strong cognitive link between conjecture production and proof construction. A detailed description is given of this process and of how it surfaced in a teaching experiment organized by us.The conditions are discussed that may have allowed the extensive implementation of the process in the classroom. Some educational implications are sketched.
© Left to authors

1. Introduction

The purpose of this report is the introduction and justification (on an experimental basis) of a hypothesis concerning mental processes underlying the production of statements and proofs by VIII grade students.

The hypothesis stems from previous research on the feasibility of a constructive approach to theorems by students. In particular, during a teaching experiment concerning arithmetic theorems students were engaged in the production and proof of conjectures. It was observed that students kept a keen coherence between the text of the statement produced by them and the proof constructed to justify it (see Garuti & al., 1995). This textual coherence brought forward the problem of a possible cognitive continuity between the statement production process and the proving process.

The hypothesis forming the subject matter of this report is that the majority of grade VIII students can produce theorems (conjectures and proofs) if they are placed in a condition so as to implement a process with the following characteristics:

- during the production of the conjecture, the student progressively works out his/her statement through an intense argumentative activity functionally intermingling with the justification of the plausibility of his/her choices;

- during the subsequent statement proving stage, the student links up with this process in a coherent way, organising some of the justifications ("arguments") produced during the construction of the statement according to a logical chain.

Despite the undeniable differences between "deductive organization of thinking" and "argumentative organization of thinking" (Duval, 1991), we want to stress some aspects of continuity, concerning the production, during the construction of the conjecture, of the elements ("arguments") that are used later during the construction of the proof.

The hypothesis featuring as subject matter of this report, which concerns the holistic character of the theorem production, if validated and thoroughly investigated by other studies, might have important didactic consequences as to the school approach to theorems, radically calling into question the teaching traditions (see Discussion).

2. References to history and research in mathematics education

The history of mathematics shows remarkable similarities between the holistic way of producing theorems by the student, described in our hypothesis and the way of producing theorems by mathematicians: despite important differences (as to reasoning, cultural experience, institutional bonds, etc. - see Hanna & Jahnke, 1993), we can detect the existence of common features, in particular as to the intermingling between the progressive focusing of the statement and the argumentative activity aimed at justifying its plausibility. At times, in the case of the history of mathematics, this is a long process, that involves many people for many years (cf Lakatos, 1976); at times it is a personal process, traces of which are found in the notes or memories of one mathematician (cf Alibert & Thomas, 1991).

With reference to the theoretical approach to "hypotheses" proposed in Boero & al. (1995), the production of a conjecture as described by us in the Introduction can be considered as a "hypothesis" production act: that is to say, it can consist of the argumented selection (prompted by a given question made by the student himself or by others) among possible alternatives, with a margin of uncertainty, as to its validity, that can be solved through the systematically organised reasoning or a counterexample ("verification" of the "hypothesis").

In the research produced in Maher (1995), in a problem solving situation implying the necessity of formulating and justifying conjectures, a behaviour similar to the one described in this report is observed in very young students (grade IV).

All these elements prompted us to examine all over again the studies on the mathematical proof within the mathematics education research, which, on the contrary, above all point out the elements of difference between argumentative reasoning and deductive reasoning (Balacheff, 1988; De Villiers, 1991; Duval,1991; Hanna & Jahnke, 1993; Moore, 1993; Tall, 1995). It seems to us that the existence of differences, epistemological obstacles, etc. is not incompatible with the fact that students can construct the proof using elements come up during the argumentation that accompained the conjecture construction process. But every element of continuity implies the risk for students to identify processes of different nature (cf Duval, 1991). These reflections were helpful to us for the planning of our teaching experiment and for the analysis of students' behaviours: in particular:

- at the stage of construction of the teaching experiment we tried to create favourable conditions for the appearance of the cognitive unity assumed by us, but also for the spacing out by students of the conjecture production stage from the proving stage, insisting in particular on the reasons for the necessity of proof as "proof of the statement truth";

- in the analysis of protocols we tried to catch the signs of attained change in students between the perspective of the argumentation to construct the conjecture and the persuasion of its plausibility, and the perspective of its proof.

3. Description of the teaching experiment

The main difficulty which we had to face was that of finding experimental confirmation for our hypothesis. It was necessary, in particular, to create an experimentation and observation context suitable to "reveal" the nature of processes of statements and proofs production and verify the potentiality conjectured by us. Underlining indicates some crucial points.

The teaching experiment was carried out in two grade VIII classes of 20 and 16 students, at the beginning of the third school year with the same teacher. Students had already interiorized the habit of producing argumented hypotheses in different domains (mathematical and non-mathematical), writing down their reasoning. Students had already experienced situations of statements production in arithmetic and geometry; they had approached proof production in the arithmetic field (see Boero & Garuti, 1994; Garuti & al., 1995).

The task concerning the production and proof of a conjecture was contextualized in the "field of experience"(Boero & al, 1995) of sunshadows. Students had already performed about 80 hours of classroom work in this field of experience. They had observed and carefully recorded the sunshadows phenomenon over the year (in different days) and over the morning of some days. They had approached geometrical modeling of sunshadows and solved problems concerning the height of inaccessible objects through their sunshadows.

The field of experience of sunshadows was chosen because it offers the possibility of producing, in open problem solving situations, conjectures which are meaningful from a space geometry point of view, not easy to be proved and without the possibility of substituting proof with the realization of drawings.

In the two classes the activities were organised according to the following stages (whole amount of time for classroom work: about 10 hours):

a) Setting the problem

"In the past years we observed that the shadows of two vertical sticks on the horizontal ground are always parallel. What can be said of the parallelism of shadows in the case of a vertical stick and an oblique stick? Can shadows be parallel? At times? When? Always? Never? Formulate your conjecture as a general statement."

(Individual work or work in pairs, as chosen by the students)

Some thin, long sticks and three polystyrene platforms were handed, in order to support the dynamic exploration process of the problem situation.

b) Producing conjectures

many students started to work with the thin sticks or with pencils. They started to move the sticks or to move themselves to see what happened. Other students closed their eyes. The absence of sunlight or spotlight in the classroom hindered the experimental verification of conjectures they were formulating: it was the mind's eyes that were "looking". Students individually wrote down their conjectures.

c) Discussing conjectures

the conjectures were discussed, with the help of the teacher, until statements of correct conjectures were collectively obtained which reflected the different approaches to the problem by the students.

d) Arranging statements

through different discussions, under the guidance of the teacher, the following statements,"cleaned" from metaphors and more precise from a linguistic point of view than those produced by students at the beginning, were collectively attained:

-" If sun rays belong to the vertical plane of the oblique stick, shadows are parallel."

- "If the oblique stick moves along a vertical plane containing sun rays, then shadows are parallel."

- "The shadows of the two sticks will be parallel only if the vertical plane of the oblique stick contains sun rays."

The first two statements stand for two different ways of approaching the problem on the part of the students: the movement of the Sun and the movement of the sticks; the third statement makes explicit the uniqueness of the situation in which shadows are parallel.

After further discussion the collective construction of the two statements below was attained:

- "If sun rays belong to the vertical plane of the oblique stick, shadows are parallel. Shadows are parallel only if sun rays belong to the vertical plane of the oblique stick "

- "If the oblique stick is on a vertical plane containing sun rays, shadows are parallel. Shadows are parallel only if the oblique stick is on a vertical plane containing sun rays"

In order to help the students in the proving stage it was preferred not to express the statement in its standard, compact mathematical form "if and only if..." (its meaning in common Italian cannot be distinguished from the meaning of "only if..." ) .

e) Preparing proof; the following activities were performed

- individual search for analogies and differences between one's own initial conjecture and the three "cleaned" statements considered during the stage d);

- individual task: "What do you think about the possibility of testing our conjectures by experiment?"

- discussion concerning students' answers to the preceding question. During the discussion, gradually students realize that an experimental testing is "very difficult", because one should check what happens "in all the infinite positions of the sun and in all the infinite positions of the sticks".

This long stage of activity (about 3 hours) was planned in order to enhance students' critical detachment from statements, motivate them to proving and make clear that since then classroom work would have concerned the validity of the statement "in general".

f) Proving that the condition is sufficient (activity in pairs, followed by the individual wording of the proof text);

g) Proving that the condition is necessary (short discussion guided by the teacher, followed by the individual wording of the proof text).

h) Final discussion, followed by an individual report about the whole activity (at home).

The following materials were collected: videotapes of the initial stages (a and b); tape-records of discussions and teacher-students interactions; all the students' individual written texts.The data which we are about to consider mainly concern stages b) and f).

4. Students' behaviour

All students actively took part in the production of the initial conjecture. 29 students (over 36) were able to follow the following activities (from c to h) in a productive way.

For each type of students' behaviour one example of written texts individually produced by students during the stages b) and f) will be reported entirely. At this stage of the research we deem important to dwell on typical behaviour that can justify the plausibility of our hypothesis and to examine it more deeply (in view of its subsequent and more extensive confirmation).

It is possible to see how at the conjecture formulation stage there is much inaccuracy from the point of view of language, concerning in particular the expressions used to indicate a vertical plane containing sunrays. Through gestures with the hands or the movement of sticks it is clear that the students intend to indicate a vertical plane, but often they call it "direction of rays". During the experiment this inaccuracy is gradually overcome: "concepts in act" (Vergnaud) receive appropriate names. Another aspect concerns the terms "it can be seen", "looking" (referred to shadows): it is worthwhile to remember that no sunlight or spotlight was available in the class, therefore the students looked and saw with their imagination.

4.1. Correct conjecture with justification (21 students)

Underlining indicates traces of connections between conjecture production and proof construction.

Formulation of the conjecture with shifting of the stick:

(Beatrice) "I tried to put one stick straight and the other in many positions (right, left, back, front) and with a ruler I tried to create the parallel rays. I sketched the shadows on a sheet of paper and I saw that: if the stick moves right or left shadows are not parallel; if the stick is moved forward and back shadows are parallel. Shifting the stick along the vertical plane, forward and back, the two sticks are always on the same direction, that is to say they meet the rays in the same way, therefore shadows are parallel. Whereas shifting the stick right and left the two sticks are not on the same direction anymore and therefore do not meet the sun rays in the same way and shadows in this case are not parallel. Shadows are parallel if the oblique stick is moved forward and back in the direction of sunrays."

Proof: "Shadows are parallel because, as we already said, sun rays belong to the vertical plane of the oblique stick.

But all this does not explain to us why this is true. First of all, though the sticks stand one in an oblique and the other in a vertical position, they are aligned in the same way and if the oblique stick is moved along its vertical plane and is left in the point in which it becomes vertical itself we see that they are parallel and, as a consequence, their shadows must naturally be also parallel, and also parallel with the shadow of the oblique stick, which has the same direction of that produced by the imaginary, vertical stick."

In this case the justification produced at the beginning("meet the sun rays in the same way") is the one that in the following proof makes Beatrice imagine the oblique stick moving along the vertical plane containing sun rays.

Formulation of the conjecture with the movement of the Sun

(Sara) "They could be parallel if I imagine to be the sun that sees and I must place myself in the position so as to see two parallel sticks. In this way the sun sends its parallel rays to enlighten the sticks. But if the sun changes its position it will not see the parallel sticks and, therefore, their shadows will not be parallel either. Shadows can be parallel if the oblique stick is on the same vertical plane as the sun rays."

Proof: "If the sun sees the straight stick and the oblique stick parallel it is as if there were another vertical stick at the base of the oblique stick. If this stick is in front of the oblique stick its shadow covers the shadow of the oblique stick. These shadows are on the same line, therefore, the oblique and vertical sticks shadows are parallel."

In this case the initial idea "I imagine to be the sun" seems to suggest the main argument of the proof (the shadow of the imaginary, vertical stick covers the shadow of the oblique stick).

Concerning production of the statement, Beatrice's and Sara's texts give evidence of complex mental processes correspondig to our hypothesis.

Concerning proof, both texts show interesting traces of the detachement from the problem situation (e.g.: "I imagine to be..." becomes "If the sun sees" ) and the original statement. Students seem to be aware that it is necessary to validate the statement by a reasoning process ("But all this does not explain to us why this is true." ). Many other texts show similar aspects.

We notice that in both cases above, just as for the majority of students, the dynamic process that brought to the production of the statement (movement of the sun or movement of the stick) is found again in the proving process. Yet the dynamic exploration implemented during the construction of the proof, though it shows remarkable similarities with the one implemented during the production of the conjecture as to the type of movement, differs deeply as to the function assumed in the thinking process: from a support to the selection and the specification of the conjecture, to a support for the implementation of a logical connection between the property assumed as true ("vertical sticks produce parallel shadows" ) and the property to be validated.

4.2. Correct conjecture without justification (6 students)

6 students out of 36, be their level high or low, formulated the conjecture correctly, but during the formulation did not manage to produce arguments backing up their hypothesis. This fact seems somehow to affect the subsequent proof that turns out to be lacking in "arguments" and rather confused.

(Elisabetta) Conjecture: "In some cases, although the oblique stick is in a position different from that of the vertical stick, the parallelism is kept, whereas in other cases the parallelism in shadows is not kept. Therefore, shadows can be parallel only if the oblique stick [meaning with a gesture the vertical plane] is parallel to the direction of the straight stick shadow, that is to the sun rays."

Proof: "Our statement is true because if the vertical plane of the oblique stick gathers the sun rays as that of the vertical stick, then the two shadows will be projected on the same line."

4.3. Wrong conjecture (9 students)

9 students, be their level high or low, produce wrong conjectures probably suggested by the principle"sun rays are parallel, then ..." or by drawings that owing to their bidimensional nature may be misleading, and are also static and so they may stick the attention on particular situations.

(Vincenzo) Conjecture: "In my opinion shadows cannot be parallel if the two sticks are one vertical and the other not vertical. I took the two sticks, I put them in a vertical position and shadows were parallel, then slowly I moved the right-hand side stick and noticed that its shadow moved. In my opinion they do not remain parallel, because if I have two vertical sticks, their shadows are parallel because rays are parallel, that is to say they come across the obstacle and form the shadow. But if I move slowly, rays that were hindered before now pass by, though they are hindered from another point, that is to say the shadow moves and, therefore, it is not parallel anymore."

At the proving stage, after classroom discussions, 6 of these students "make up for" the lost grounds and it can be noticed how their proof is full of constructions and argumentations, as if these students had to reconstruct the conjecture to be proved:

(Vincenzo) Proof: "The statement is true because: let us imagine to have an oblique stick and a vertical stick. Let us imagine to draw an imaginary line, perpendicular to the horizontal plane, starting from the point of the oblique stick. Let us do the same thing with the vertical stick but the other way round, meaning that I draw an imaginary oblique line parallel to the oblique stick.

It happens that I get two vertical lines with parallel shadows and two oblique lines with parallel shadows. The imaginary stick casts a shadow into the direction of the oblique stick, as a consequence the shadows between the oblique stick and the vertical stick are parallel".

5. Conclusions

It appears to us that the data just illustrated are consistent and make our hypothesis plausible.

Actually, as concerns the production of the statement, argumentative reasoning fulfils a crucial function: it allows students to consciously explore different alternatives, to progressively specify the statement and to justify the plausibility of the produced conjecture (see 4.1.). On the other hand, students that produced wrong conjectures later show the need of reconstructing the valid conjecture in order to produce the proof (see 4.3.).

The fact that poor argumentation during the production of the statement always corresponds to lack of arguments during the construction of the proof seem to confirm the close connection that exists between production of the conjecture and construction of the proof (see 4.2.).

Moreover, the consistency among personal arguments provided during the production of statements and the ways of reasoning developed during the proof seems to be confirmed:

- by the fact that the type of argumentative reasoning made during the production of the statement by one student is resumed by him/her (often also with similar linguistic expressions) in the justification of the statement subject to proof;

- by the fact that the kind of dynamic process (movement of the sun or the stick) recorded at the conjecture stage is almost always the same as the one used at the proof stage.

A further element surfaces during the teaching experiment: it can be observed that at the statement formulation stage the exploration by students almost always concerns both the parallelism and the non-parallelism, even if this process is not "abridged" (obviously, owing to the lack of experience in standard mathematical formulation) in a statement such as "if and only if".

6. Discussion

As mentioned in the introduction, the hypothesis on which we worked seems to have important didactic implications, since it calls into question the traditional school approach to theorems.In fact, usually in Italy and in other Countries the teacher asks the students to understand and repeat proofs of statements supplied by him, which appears one of the most difficult and selective tasks for grade IX-X students. Only as possible last stage (often reserved to the top level students or students choosing an advanced mathematical curriculum) the teacher asks the students to prove statements, generally not produced by students but suggested by the teacher. Even more seldom students are asked to produce conjectures themselves. If our hypothesis is valid, during this traditional path students' difficulties can at least partly depend on the fact that they should reconstruct the cognitive complexity of a process in which mental acts of different nature functionally intermingle starting from tasks that by their nature bring them to partial activities that are difficult to reassemble in a single whole. Our teaching experiment suggests an alternative didactic path.

Just for the importance of such didactic implications we deem opportune to critically analyse some possible limits of the study made so far and to sketch further developments of it.

6.1. Critical analysis of findings and further research

First of all, we must consider in what sense students have performed a mathematical activity concerning theorems.

The object of the experiment is a hypothesis concerning the physical phenomenon of sunshadows; it has as a geometric counterpart, at the level of model, a statement of parallel projection geometry. Students produce their conjecture as a hypothesis concerning the phenomenon of sunshadows; when they verify their conjecture most of them seem to be aware of the fact that they must get the truth of the statement by reasoning, starting from true facts. Most of them produce a validation realized through a deductive reasoning. Actually their reasoning starts from properties considered as true ("two vertical sticks produce parallel shadows") and gets the truth of the statement in the "scenary" determined by the hypothesis.

In this way, students produce neither a statement of geometry "strictu sensu", nor a formal proof: objects are not yet geometric entities, deduction is not yet formal derivation. But their deductive reasoning shares some crucial aspects with the construction of a mathematical proof. Moreover, the whole activity performed by students shares many aspects with mathematicians' work when they produce conjectures and proofs in some mathematics fields (e. g.: differential geometry): mental images of concrete models are frequently used during those activities. As to proof, mathematicians frequently come near to realize the ideal of the formal proof only during the final stage of proof writing. During the stage of proof construction, the search for "arguments" to be "set in chain" in a deductive way is frequently performed through heuristics, the reference to analogical models and keeping into account the semantics of considered propositions (cf Alibert & Thomas, 1991).

For these reasons we think that the activity performed during our teaching experiment may represent an approach to mathematics theorems which is correct and meaningful from the cultural point of view.

In our opinion, the continuity aspects hihglighted by us represent a huge potentiality for the development of the students' ability to prove conjectures; nevertheless, this potentiality needs an adequate educational context in order to surface successfully. In planning our teaching experiment we singled out some conditions that are probably necessary to this end; they concern:

- the didactic contract set up in the classroom (the production of a conjecture to solve an open problem, the value of an hypothesis as an "argumented choice");

- the didactic path in which the task is inserted (particularly, in our case, the choice of the field of experience of sunshadows as a long term learning environment);

- the management of classroom work after the task (individual activities alternating with activities in pairs and discussions; activities to prepare the proof stage - see e).

We are not as yet able to establish whether all the conditions that we singled out are actually necessary and sufficient for the extensive implementation of the process that we recorded in our teaching experiment.

It is necessary to ascertain what the actual weight of the didactic contract is, through comparisons with classes having a different history behind.

It is necessary to find out how much, and how, the cognitive unity of theorems appears also in mathematical fields other than geometry (and, in particular, that of "shadows geometry").

It appears also important to ascertain the consequences of "theorems cognitive unity" experiences on the activity of standard theorems proving, proposed through their statements.

Finally, it seems opportune to investigate the connections, the analogies and the differences between the procedures for the dynamic exploration of the problem solving situation during the production of the conjecture and, during the process of proof construction, the procedures for the dynamic exploration of the situation determined by the hypothesis.

Acknowledgements. Carlo Dapueto and Pier Luigi Ferrari helped us to clarify and develop some ideas of this paper. We thank them very much.

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