Garuti R., Boero P., Lemut E. (1998)
Cognitive unity of theorems and difficulty of proof.

1. Introduction

In preceding papers regarding the approach to geometry theorems in school (Garuti & al,1996; Mariotti & al,1997), a specific theoretical construct ('cognitive unity of a theorem') was introduced in order to stress the importance of a holistic approach to theorems and to interpret some of the difficulties met by students in the traditional approach to proof. Cognitive unity of a theorem is based on the continuity existing between the production of a conjecture and the possible construction of its proof.

The idea of this construct initially came from the epistemological analysis of work done by past and present geometers, which revealed many examples of continuity between the production of a statement and the construction of its proof, in particular as concerns the relationship between, on the one hand, specifying the objects of the conjecture, determining stricter hypotheses or stating a new weaker conjecture and, on the other, performing trials to prove the statement (Lakatos,1976; Thurston, 1994). We then found a cognitive counterpart of this analysis: in a teaching experiment concerning the production of theorems (conjectures and proofs) by beginners in a mathematical modelling problem situation concerning sunshadows, we found experimental evidence of the cognitive unity between the phases of conjecture production and proof construction. We expressed this unity in the following terms:

(CU) "during the production of the conjecture, the student progressively works out his/her statement through an intensive argumentative activity functionally intermingled 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 previously produced arguments according to a logical chain.".

These findings led us to state that in order to bring about a smooth approach to theorems in school, it is necessary to consider the connection between conjecturing and proving, in spite of the undeniable differences between these processes. We also wondered whether in a traditional school approach to theorems one of the difficulties that students face could be that of reconstructing that unity, when it happens to be hindered. A very particular, we should say extreme, situation is that of the task "prove that ..." : in this case the process of conjecturing is not demanded and the unity is broken. Unity may be reconstructed only by the riappropriation of the statement through a new process of exploration, i. e. reconstructing the whole cycle: exploring, producing a conjecture, coming back to the exploration, reorganizing it into a proof.

These reflections suggested the following questions: Can the cognitive unity construct be a tool allowing teachers and researchers to predict and interpret students' difficulties when they have to prove a given statement? Can it be a tool that allows the teacher to select appropriate tasks that increase in difficulty, in relation to the increasing difficulty in establishing continuity between the statement and the proving process? The study reported in this paper aims to produce some partial ansvers to these questions.

2. Towards a Tool for Interpreting and Predicting Difficulties in Proving

In the aforementioned teaching experiment, we analysed the behaviour of all the students who had produced mistaken conjectures. When they had to prove the statement finally agreed on in classroom discussions, they were in the same situation as the students who have to prove a statement they themselves had not produced (that is, in the same situation as the traditional school approach to proof -see 1.). We discovered that all the students who had successfully managed the proving activity appropriated the conjecture to be validated through a dynamic exploration of the problem situation; they produced arguments for the conjecture's plausibility, which then were useful for proving the statement (a process similar to the production and validation of an original conjecture, according to the analysis performed by Boero & al, 1995). The activity of dynamic exploration and search for arguments for plausibility appeared to be a necessary step towards the construction of the proof.

Research work by Simon (1996) and Harel (1996) suggested that the students' proving processes should be considered from another wider perspective. Simon describes the possible role of "transformational reasoning that involves envisioning the transformation of a mathematical situation and the results of that transformation" in "theorem generation, making of connections among mathematical ideas and validation of mathematical ideas". Harel points out the role of "transformational proof schemes" as "foundation for all theoretical proof schemes". Thus, Harel's and Simon's work suggested us to take into account the different kinds of transformation (of what? in relationship with what?) that can intervene during the proving process. Indeed, we may consider different objects and different levels of transformation: a purely syntactic, but goal oriented transformation (like in some proving processes based on algebraic transformations - see Boero, 1997); a transformation of the situation represented by the statement, in order to generate another statement, easier to prove; a translation into another language (for instance, from verbal to algebraic language), etc. "Dynamic explorations" considered by Boero & al. (1996) are based on suitable imagined, or concretely performed transformations of space configurations. We also note that a proof (the final product of an effective proving process) may be regarded, by itself, as a chain of transformations of the statement according to logical rules. Further possibilities of "transformational reasoning" in proving are suggested by Polya's work concerning problem solving, combined with the idea of "mathematical theorem" as a statement, its proof and the reference theory (see Mariotti & al, 1997). Polya pointed out that in some cases it is very profitable to perform the global "transformation of the problem" to be solved by setting it in a theory different from that in which the problem was originally conceived. As an example of transformational reasoning in proving applied to the reference theory, let us consider the history of the proof of the Fermat's last theorem. In this case exploration of the statement in the field of arithmetics did not produce arguments which might be immediately exploited to construct the proof; construction of the proof called for a major transformation of the statement (by interpreting it as a particular case of a conjecture concerning a different field of mathematics- a change of reference theory) together with a jump in the complexity of the elaboration of appropriate arguments.

Taking into account these reflexions and our preceding work, finally we reconsidered our theoretical construct of the cognitive unity of a theorem (see CU) in order to get a pointer of the difficulty of proving a given statement and, consequently, as a tool to predict the level of difficulties met by students. We defined as gap between the exploration of the statement and the proving process the distance between the arguments for the plausibility of the conjecture, produced during the exploration of the statement, and the arguments which can be exploited during the proving process. In some cases the gap remains inside a reference theory; in other cases (see Fermat's theorem), the gap concerns also the transition from arguments in one reference theory to arguments in another reference theory. At this stage of our research, we may formulate the following tentative hypothesis:

the greater is the gap between the exploration needed to appropriate the statement and the proving process, the greater is the difficulty of the proving process.

Taking into account the preceding reflections, we may say that this gap may be reduced through suitable transformations (concerning the formulation of the statement, and/or the situation represented by it, and/or the reference theory, etc.); the exploration of the situation described by the statement and these transformations appear as necessary ingredients of the construction of the proof of a given statement.

We shall present some examples that illustrate and support the validity of this perspective in the case of proofs of given statements which do not need changes concerning the reference theory. These were chosen in a non-geometrical field (the elementary theory of numbers) in order to avoid our perspective, which was elaborated in the geometrical domain, being regarded as context specific.

In spite of all the experimental evidence collected till now, we think that an important work remains to be performed, i.e. the final, precise formulation of our hypothesis, its verification, its integration in the perspective outlined above (where the idea of "transformation" in interplay with the idea of "dynamic exploration" plays a major role) as well as the implementation of its educational implications. These topics will be outlined in Section 5.

3. Construction of Proofs of Given Statements: Some Examples

The following examples were produced by students ranging from grade VII (lower-secondary school) to undergraduate level. They have been chosen as representative cases, so their comments point out general aspects. All the examples concern students accustomed to dealing with open problem situations and reporting in detail their reasoning in written form. All the tasks are of the type "Prove that..."

3.1. "Prove that the sum of two consecutive odd numbers is a multiple of 4 "

Two examples of proof are reported:

a) a grade X student

" I can write two consecutive odd numbers as 2k+1 and 2k+3, so I will find: (2k+1) +(2k+3) = 2k+1+2k+3 = 4k+4 = 4(k+1). The number I get is a multiple of 4".

This student quickly appropriated the statement; he wrote down the sum of the two consecutive odd numbers in a suitable way (by a translation from verbal language into algebraic language) and then performed suitable standard algebraic transformations; the interpretation of the final formula validated the statement.

b) a grade VII student

" I shall perform some tests: 3+5=8 ; 1+3=4 ; 5+7=12; then I can write these additions in this way: 3+5=3+1+5-1=4+4=8 (the same for the other additions). It is like adding the even number in the middle position to itself, and the double of an even number is always a multiple of four".

In this case the student does not know algebraic language, so he needs to explore the statement in order to transform it. The exploration shows the equivalence between adding two consecutive odd numbers and adding two appropriately chosen equal even numbers. The interpretation of the result of the performed transformation ("double of an even number..") allows the validation of the statement. We may remark that the student moves inside the frame of a very elementary theory of numbers as "reference theory" (appropriately exploiting properties like "The sum of two even numbers is even").

In spite of the differences between the two proving processes (usage of algebraic language vs natural language), in both cases there is continuity between the appropriation of the statement and the construction of the proof, and the transformation of the statement is not a difficult task.

3.2. "Prove that the number (p-1)(q² -1)/8 is an even number, when p and q are odd numbers " (following an idea by Arzarello, 1993).

The proof chosen as an example was produced by a fourth year university student (in a mathematics education course concerning problem solving). In order to facilitate the analysis, the student's text is subdivided into "episodes":

Ep. 1: " p and q are odd, then p=2m+1 and q=2n+1".

Ep. 2: " I shall analyse the formula: (p-1) is even, q2 is odd, then (q2-1) is even, so (p-1)(q2-1) is even, being a product of even numbers. But in this way I get no result because in general it is not true that an even number divided by another even number makes an even number.

Ep. 3: " I shall try a transformation:

(2n+1-1)[(2m+1)2-1]/8 = 2n(4m²+4m+1-1)/8 = 2n 4 (m²+m)/8"

Ep. 4:" if p=1 e q=3 then 0*8/8=0; p=5 and q=7 then 4*48/8=24; p=11 and q=13 then 10*168/8=210. It seems that by substituting q with an odd number, q2-1 is always divisible by 8. If I succeed in proving this in general, everything is fine, because at this point I would get an integer number multiplied by an even number (that is, p-1) and so it is obvious that the result is even.";

Ep. 5: " Now I shall prove that, if q is odd, q²-1 is always a multiple of 8; q=2n+1 then q²-1= 4n²+4n+1-1= 4n(n+1). This is at least divisible by 4, and so what remains is n(n+1), which is surely divisible by two, because if n is even everything is fine, if n is odd then (n+1) is even. We may conclude that q²-1 is a multiple of 8";

Ep. 6: "I know that (p-1)(q²-1)/8 is even if p and q are odd. The conclusion arrives quickly after the illumination that q²-1 is divisible by 8". [our underlining].

Analyzing this student's performance, we may remark that:

• The first episodes (1, 2 and 3) apparently lead nowhere, but they serve as non-goal-oriented exploration of the statement. It is as if the student is "testing the ground" to find something, but without knowing what. The meaningful passage appears in Episode 4, and the manner in which it arises is typical of conjecturing: numerical tests, observation of a regularity which leads to a conjecture. The underlined sentence is illuminating: it is rather frequent, during exploration of a statement, to arrive to this point: "if I could prove B, then I would have proved A" (crucial lemmas are frequently generated in this way). From this moment on, students' operations are goal-oriented and intended-anticipatory (see Harel, 1996); that is, they aim "to derive relevant information that deepens one's understanding of the conjecture and potentially leads to its proof or refutation"
• Exploration of the statement leads the student to generate a new theorem ("q²-1 is a multiple of 8, if q is odd"). This aspect, which we consider particularly interesting, confirms what Simon (1996) observed, although in different situations, about trasformational reasoning (see 2., quotation from Simon).
• Between Ep. 4 and Ep. 5 we may observe, from the student's subjective point of view, a change in the status of the sentence: "(q²-1) is divisible by 8". Indeed, at the beginning it is considered as a conjecture; the student is not sure about its truth and writes "It seems that"; it then becomes a statement to prove: the student starts Ep. 5 by writing "I shall prove that". The same behaviour had been observed also in a very different situation, with 8th-graders (see Garuti & al, 1996).

In general, we may say that this student appropriates the statement by transforming it and establishing continuity between the exploration of the transformed statement and the proving process. Metaphorically we may say that through the exploration of the statement the student tries to unravel a tangle, and then by following the thread builds up the web of proof. In our opinion in the case of this theorem the gap is greater than in the preceding case, because the appropriation of the statement needs a more complex transformation and finding the "thread" (which allows continuity) is more difficult. In order to support this hypothesis, we may consider the behaviour of students who fail to costruct the proof. Some of them meet difficulties in interpreting the same formula reported in Episode 3, which was obtained by them through standard algebraic transformations; some of them (in episodes similar to Episode 2) think they have proved the statement by writing that the product of (p-1) and (q²-1) is always an even number

3.3. "Prove that if two numbers are prime to one another, the sum will also be prime to each of them " (Euclid's Elements, Book 7, Prop. 28; taken from Heath, 1956).

Also in this case, the chosen example concerns a fourth year mathematics student.

Ep.1 :" GCD(a,b)=1 then GCD(a,a+b)=1 and GCD(b,a+b)=1
I shall try to reason by contradiction: if GCD(a,a+b)=c with c‚1, then a+b=cn, consequently (a+b)/c=n, that is a/c+ b/c=n
I can say nothing because for instance 1/2+1/2=1, but 1 is not divisible by 2";

Ep.2 :" As a is divisible by c and c=GCD(a,a+b), it follows that c divides both a and a+b. Then a/c=m and then b/c=n-m, that is c divides b: absurd!"

Ep.3: " I think that this is true; I shall try to formalize it better:

GCD(a,a+b)=1 with c‚1; then a+b=cn, a=cm; [*]

(a+b)/c=n , a/c+b/c=n , m+b/c=n ; b/c=n-m=m' then b= cm'

a and b have at least c as a common divisor, but then GCD(a,b)‚1 and this is absurd. The idea of reasoning by contradiction came to mind because I consider it natural when I have to prove things of this kind.".

Analysing this proving process we may note that:

• In Ep. 1 the student translates the statement into symbolic language and transforms "to be prime to one another" into GCD(...)=1, then performs standard algebraic transformations leading him nowhere; note how his interpretation of "GCD(a,a+b)=c with c‚1" in terms of "a+b=cn" is only partial!
• Full appropriation of the statement happens in Ep. 2, when the student interprets the property formally expressed by "c=GCD(a,a+b) " as "c divides both a and a+b", and then translates this statement into formulas which allow an easy and effective algebraic transformation. This very passage will allow him to prove the statement by continuity with the preceding exploration;
• The passage marked with [*] is the missing link in Ep. 1; this passage becomes explicit only after the exploration performed in the Ep. 2;
• Performing a proof by contradiction presents no difficulty for this student as for other students in the same group.

In general, we may remark that exploration of the statement is made difficult by the fact that the statement encapsulates a non-trivial "logical" content. The gap between the exploration of the statement and the proving process is relevant: exploration may remain at the level of formal transformations of the statement (for instance, from "to be prime to one another" to "GCD(...)=1" to a contradiction: "if GCD(...)=c with c‚1" ) without fully penetrating the meaning of the statement. On the contrary, in the preceding case 3.2. an effective exploration was possible through standard algebraic transformations or conjectures about numerical cases.

Once again, our interpretative hypothesis is confirmed by students who failed the proof: they made numerical trials (which provide arguments for plausibility), they performed formal transformations (like at the beginning of Ep. 1), but they did not succeed in penetrating the logical knots of the statement; for instance, some of them, while reasoning by contradiction, erroneously supposed that "a is a multiple of b "(or "b is a multiple of a"), failing the interpretation of "to have common divisors".

4. Returning to a Preceding Experiment: A Deeper Interpretation

Our hypothesis concerning the cognitive unity of a theorem allows a deeper interpretation of what happened in a preceding teaching experiment, described in Boero & al. (1995). Seventh grade students had produced (through exploration of numerical examples) two different formulations of the same property:

a) "A number and the number immediately after have no common divisors except for the number 1" (We called this a "relational statement").

b) "If you add 1 to a number, all its divisors change, except 1". (We called this a "procedural statement")

We observed how in this particular case the different formulations of the statement influenced the proving process: students who referred to the relational statement were not able to go beyond exploration of the conjecture. Indeed they considered, in some numerical cases, the divisors of a number and the divisors of the following number, observing that there was no common divisor, with the exception of 1. No general proof was constructed.

On the contrary, some students referring to the procedural statement were able to construct a proof; they considered the divisors of a given number, then they transformed it into the following number and checked if the divisors of the first number divided also the second, discovering that the added unit constituted the remainder of the division of the increased number by the divisors of the initial number (and so they developed an appropriate, general argument for a proof).

These different behaviours led us to hypothetize the existence of a "textual continuity" between the statement and the proof. We now believe we can interpret those students' behaviours in a deeper and more appropriate way: the gap between the exploration of the statement and the proving process is less with the second formulation (the exploration provides a suitable, crucial argument for the proof).

5. Concluding Remarks and Further Developments

At this stage of the research we can say that, from the educational point of view, the teacher can use the construct of the cognitive unity as a tool for predicting and analysing some difficulties met by students when they have to construct a proof. In particular, the way a statement supplied by the teacher is formulated is of relevance, especially for beginners (see 4.). But it is also important that students gradually learn to transform autonomously the given statement in order to establish a continuity between exploration of the statement and construction of proof (indeed, in an example like that discussed in Section 4. students can obtain an easy proof by transforming the statement). This remark confirms the importance of transformational reasoning and the necessity of nurturing it (Simon, 1996, p. 207). The problem of how to implement this indication in class work remains still open!

From the research point of view, let us consider the case of proofs needing a change of the reference theory: the skills needed, and especially the nature of the exploration process leading to this change, should be carefully investigated.

We also think that the study we have reported in this paper could be developed further in order to understand better the nature of the exploration of the situation described by a given statement and the conditions which allow to make a productive connection between such exploration, transformational reasoning and construction of proof. The need for further investigations is made clear in the following example.

In an Alessandria University orientation (non selective) test, students had to prove that "Each number which is even and larger than 2 can be written as the sum of two different odd numbers". In spite of the apparent ease of the task, about 90% of students were unable to produce a complete proof.

The most common approaches can be described as follows:

i) after some rather casual numerical trials, some students proved that "the sum of two different odd numbers is even"; this approach could be the effect of an effort to transform the statement, given the difficulty of proving it, with a final approach to a statement not equivalent to the original one but easier to prove!

ii) some students considered many numerical cases, without finding any regularity; in this case, the exploration remained non-goal oriented (as concerns the development of the proving process), although it confirmed the validity of the statement in many numerical cases (so providing arguments for its plausibility);

iii) other students wrote(for instance)4=1+3;6=1+5;8=1+7;10=1+9;12=1+11,but they were not able to elicit the general relation ("each even number is the sum of two odd numbers,1 and the preceding number"); indeed some of them grasped the existence of a "regularity", but wrote:"I am not able to write it in general".In few cases, the interviews recorded after the test revealed that the difficulty derived from insufficient mastery of algebraic language; in other cases, the student was not able to see that the second addendum of the sum was always odd because it preceded an even number!

These behaviours show the necessity of taking into account other aspects of the proving process, which concern both the exploration process and the transformation of the statement: the nature of the "control function" of transformations, and how to develop it (see i); why does the exploration process prove in some cases absolutely blind (see ii); the role (and the difficulty) of that particular exploration which aims to interpret the results of a transformation or a preceding exploration (see iii).

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