Boero P., Garuti R., Lemut E. (1999) About the generation of conditionality of statements and its links with proving. PME XXII proceedings. Haifa, Israel. PME XXIII, Haifa, Israel. Volume 2, pp. 137-144.
	Abstract Conditionality of statements (i.e. the fact that statements of most theorems are implicitly or explicitly shaped according to the "if A then B" clause) has been a peculiarity of theorems throughout the history of mathematics. The aim of the research partially reported in this paper is to detect and describe a set of processes of generation of conditionality in statements (PGC) that is wide enough to cover the majority of PGCs that occur in different fields of mathematics. In this paper we will describe four kinds of PGCs, along with some productive links between these PGCs and the processes of construction of proof.
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1. Introduction In our previous investigations we considered the conditionality of statements (i.e. the fact that statements of most theorems are implicitly or explicitly shaped according to the "if A then B" clause) as a peculiarity of theorems throughout the history of mathematics (see Boero and Garuti, 1994). We also considered two possible ways of generating conditionality in the geometry field, and posed the problem of finding other ways (see Boero et al, 1996). In one case (that of generation of conditionality by a "time section" in the exploration of the problem situation), a strong link was detected in students' protocols between the process of generation of conditionality (PGC) and the process of construction of proof (see Garuti et al, 1996).    The aim of the research reported in this paper is mainly to detect a sufficiently wide set of PGCs and describe them in order to cover the majority of PGCs that occur in different fields of mathematics. We will describe four kinds of PGCs; we may add that no other PGC was detected in the examined protocols (see 4.1). In addition, as part of our continuing research on the cognitive unity of theorems (see Garuti et al, 1998), we will here describe some productive links between the PGCs and the processes of construction of proof (see 4.3.).    This research may have important implications for mathematics education: it seems to be possible (through suitable tasks) to let students experience different kinds of PGCs that are important in mathematical activities concerning theorems (see 5.). 2. Background Research Psychology has always devoted much attention, in a developmental perspective, to reasonings concerning conditionality; a landmark contribution in this direction is given in the early scientific production of Piaget (see Piaget, 1924, Chapter 2). More recently, psycholinguistic research has explored in depth the acquisition of the "if... then..." clause, analysing its context-dependence and its links with other aspects of mental development, in particular those related to mastery of causality (see French, 1985 for a survey).    On the mathematicians' side, processes related to producing conjectures and proving theorems have for decades been a fundamental point of attention: we may quote Hadamard (1949), Polya (1962) and, recently, Thurston (1994). Progressively, this attention has shifted from descriptions of personal experiences or very general statements to more precise hypotheses.    Recently, research in the fields of logic, foundations of mathematics and artificial intelligence have converged on the need for understanding of how humans actually produce conjectures, prove theorems and exploit the knowledge thus acquired: "We do not yet see how humans are able to discover proofs, we cannot yet explain how they affect the human mind" (Robinson, 1998).    Educational research too has focused on the topic of analyzing processes of production of conjecture and construction of proof, in order to create suitable learning environments and tasks to enhance them. Recent contributions in this direction are the theoretical constructs of "transformational reasoning" by Simon (1996) and "transformational proof scheme" by Harel and Sowder (1998). According to Simon, "Transformational reasoning is the physical or mental enactement of an operation or set of operations on an object or set of objects that allows one to envision the transformations that these objects undergo and the set of results of these operations. Central to transformational reasoning is the ability to consider, not a static state, but a dynamic process by which a new state or a continuum of states are generated" [...] "[...] transformational reasoning is a natural inclination of the human learner who seeks to understand and to validate mathematical ideas. The inclination, like many other inclinations [...] must be nurtured and developed [...]." [...] "It seems that transformational reasoning can serve several cognitive functions including, theorem generation, making of connections among mathematical ideas and validation of mathematical ideas". In our research about historical-epistemological, cognitive and educational aspects of conjecturing and proving (see Boero & Garuti, 1994; Boero et al, 1996; Garuti et al, 1996; Garuti et al, 1998) conditionality of statements is a point of major concern.    Conditionality has been a crucial peculiarity of theorems throughout the history of mathematics. Heath (1956) points out how conditionality is present in Euclid's "Elements" theorems, whether in explicit terms or in implicit terms. In the latter case, the statement can be reformulated in order to make the "if A, then B" clause explicit (for instance in the case of Pythagoras' well known theorem, the usual statement "in a rectangular triangle, the square built up on the hypotenuse... etc" can be reformulated as follows: "If a triangle is rectangular, then..."). We may remark that, today, statements of theorems do not differ from Euclid's as concerns conditionality. After Hilbert's revolution the epistemological perspective has changed considerably as concerns the nature of truth expressed by the statement of a theorem, the nature of postulates, the requirements of proof. However, the formulation of a statement in (explicitly or implicitly) conditional terms remains a peculiarity of most theorems. Moreover, when we consider the conditionality of statements we do not limit ourselves to the textual property of statements. Its substantial importance in mathematical activities concerning theorems lies in the fact that the proving process keeps the "if A then B" clause as a crucial orienting reference for validating the statement. The difficulty is to match this evidence about the importance of the conditionality of statements from the epistemological point of view with a cognitive analysis of how it is generated during mathematical activity of conjecturing and how it is linked to the proving process.    Our research work on some PGCs detected in students' protocols (see Boero et al, 1996) pointed out some peculiarities of those processes, related to management of virtual time and space variables in students' "inner visual field" (Vygotskij, 1978, Chap. I). In particular, we described in the following way a particular kind of PGC detected in students' protocols: "the conditionality of the statement can be the product of a dynamic exploration of the problem situation during which the identification of a special regularity leads to a temporal section of the exploration process that will be subsequently detached from it and then "crystallize" from a logic point of view ("if..., then…)." We also found some links with the proving process (see Garuti et al, 1996).    Our next research work was aimed at finding other PGCs and determining more precise links between PGCs and proving processes. In this paper we will describe some kinds of "transformational reasonings" (Simon, 1996) that intervene in producing and proving conjectures (see 4.1.). 3. Method Following occasional hints, a systematic investigation was performed on students' protocols concerning conjecturing and proving. We considered: beginners' written protocols (grades from V to VIII), in order to explore some basic PGCs not yet influenced by known patterns and complex analytic techniques; these protocols were used as sources of ideas about possible generative processes; undergraduate mathematics students' protocols, in order to validate the definitions deriving from preceding analyses and make them more precise and content-independent. Most were written protocols but some recorded dialogues with the teacher were also considered. Students were attending mathematics education courses on problem solving in the last four years. A common production condition for all protocols was that in all cases the educational setting should stimulate students to write or orally express their thinking processes. In most cases this was done as real-time wording of their intuitions and endeavors, in others as on the spot reports about their reasoning. In the case of undergraduate students, this was done by systematically exploiting their written (or possibly recorded oral) solutions as anonymous texts to be discussed by their fellows, without any evaluation about correctness. In this way students recognized exhaustive wording of processes as a necessity in order to get interesting material for discussion. In the case of beginners, writing down reasoning was a part of the didactical contract in the classes engaged in Genoa Group Projects for primary and junior high school. The fields of mathematics involved were: elementary plane and space geometry, arithmetic (properties of natural numbers) and elementary algebra for VII-VIII graders: five tasks with more than 20 protocols for each task; mathematical analysis, euclidean geometry, algebra and theory of numbers for undergraduate mathematics: nine tasks, more than ten protocols for each task. The analysis of students' protocols was performed following these steps: first, detecting and trying to describe PGCs and their links with proving processes as they arose in single, clear protocols (see later for some examples); then, challenging the description through the comparison with other protocols (possibly by different students and in different fields of mathematics) that presented similarities as concerns PGCs and their links with proving; and subsequently improving the description in order to make it content-independent; finally, trying to establish a common style of description among the different PGCs that had been detected, trying to get an overall vision of them and show any possible relationships, symmetries, etc. among them. The reported results represent a final summary of what emerged during these analyses. They are quite complete as concerns the four detected PGCs, they are far from being exhaustive as concerns links with the proving processes. 4. Some Results 4.1. Processes of generation of conditionality In order to make the presentation easier to understand, examples will precede definitions. The examples will be given related to different fields of mathematics so that "invariant" elements are highlighted. The following kinds of PGCs were detected in the students' protocols, covering some different fields of mathematics. PGC1. For some examples concerning VIII graders, see Boero et al, 1996. Here are some others. EX.1.1.: geometry field, undergraduate students. Task: "In the euclidean environment, formulate and prove a conjecture concerning the possible existence of a minimum area among the areas of all triangles obtained by closing an angle with straight lines passing through a point on the bisecting line of the angle itself". One student draws the configuration angle/bisecting line/point on the bisecting line and then draws several straight lines passing through that point. Initially, these are sharply very inclined with respect to the bisecting line, and on the same side; then come other lines close to the perpendicular line and finally lines on the other side that strongly diverge from the perpendicular line. Afterwards, the student states: "It seems to me that the areas of the triangles decrease as they approach the position ... the perpendicular line … I see triangles growing and growing on one side without any balance on the other side." (She shades in one large triangle emerging from the isosceles triangle and the corresponding smaller incoming triangle). "Perhaps the conjecture is: if the passing through line is perpendicular to the bisecting line, the area gets its minimum". EX.1.2.: algebra, undergraduate students. Task: "Let ax+by be an expression where a and b are positive integers, x and y integers; find out under what conditions on a and b the expression ax+by can assume its minimum positive integer value.". A student writes: "Let me try: a=4 and b=6: 4*1+6*1=10; 4*2+6*1=14; 4*2+6*2=20. The results are increasing; but I can also use negative values for x and y: for instance, 4*2+6(-1)=2; 4*2+6(-2)=-4; 4*3 + 6(-2)=0; 4(-4)+6*3=2. It seems to me that the results can not go lower than 2. I try with 3 and 5: 3*1+5(-1)=-2; 3*2+5(-1)=1. I reached 1, which is the minimum positive integer value. It is easy now, perhaps because 3 and 5 do not have any common divisor (but 1). The conjecture: if a and b do not have any common divisor (but 1), the minimum value is 1." Generally speaking, a PGC1 can be described as a time section in a dynamic exploration of the problem situation: during the exploration one identifies a configuration inside which B happens, then the analysis of that configuration suggests the condition A, hence "if A, then B". PGC2. EX. 2.1.: the study reported in Boero & Garuti (1994) concerned VII-graders who had to express in general geometric terms "Thales' discovery" (i.e. the anecdote concerning the determination of the height of a pyramid by exploiting the proportionality between the heights of objects and the lengths of the shadows they cast). The following kind of reasoning was identified in 3 out of the 34 students: "The length of the shadows is proportional to the height of the sticks; sunrays are parallel. But straight lines might not be parallel. If the straight lines are parallel, the lengths of the segments cut by another two lines will be proportional" EX. 2.2.: undergraduate students. Task: "Can we always represent f(x)=sin(Ax+B) by finite linear combinations of products of integer powers of sinx and cosx?" "It seems so, by applying the trigonometric formulas. For instance, if A=2 and B=3, I can write: sin(2x+3)= sin2x.cos3+cos2x.sin3= 2sinxcosx.cos3 +(cos2x-sin2x).sin3. But A might be p also: we cannot write sinpx in that way. And also A=1/2 does not work: we would need roots. On he contrary, if A is an integer, it works". Generally speaking, a PGC2 can be described as: noticing a regularity B in a given situation, then identifying, by exploration performed through a transformation of the situation, a condition A, present in the original situation, such that B may fail to happen if A is not satisfied. PGC3. EX. 3.1.: algebra, undergraduate students. Task: "Generalize the following property: 'the sum of two consecutive odd numbers is divisible by 4'. Prove the property found". "Let me consider 3 consecutive odd numbers, e.g. 3+5+7=15 or 5+7+9=21. It seems to me that only divisibility by 3, which is the number of addenda, emerges. I shall try with 4 consecutive odd numbers: 3+5+7+9=24; 5+7+9+11=32; 1+3+5+7=16. What do 24, 32,16 have in common? They are divisible by 8. I shall try with 6 consecutive odd numbers: 1+3+5+7+9+11=36; 3+5+7+9+11+13=48. Both sums 36 and 48 are divisible by 12. If there are 4, the sum is divisible by 8. If there are 6, sum divisible by 12. If there are 2, we have seen that the sum is divisible by 4. It seems to me that what is emerging is that the sum of an even number of consecutive odd numbers is divisible by its double, i.e. by the double of the numbers of addends I am adding." EX. 3.2.: V-graders (cf. Bartolini Bussi et al., to appear). Task: "Ascertain what happens when the number of cog-wheels, engaged and arranged in a ring configuration, increases, having already found that three can not turn all together, while four do". "(One pupil draws 5 wheels and indicates the rotation direction with arrows) "5 wheels can not turn; (draws 6 wheels) 6 can turn (draws 7 wheels). It could turn with 4 but with 3 it could not. So, if the number of cog-wheels is even, they can turn. If it is odd, they can not". Generally speaking, a PGC3 can be described as a 'synthesis and generalisation' process starting with an exploration of a meaningful sample of conveniently generated examples. PGC4. EX. 4. 1.: (undergraduate students) In the task presented in EX. 3.1., a student begins considering 8 consecutive odd numbers and finds out that the sum is divisible by 16. He writes: "It may be that the double of how many numbers I am adding is influential in someway, but it might depend on the fact that 8 is a power of 2". He considers ten consecutive odd numbers, and finds out that their sum is divisible by 20. He concludes conjecturing that: "If n is even, the sum of n consecutive odd numbers is divisible by 2n". EX.4.2.: (V graders) In the task presented in EX.3.2. a student acts as follows: He draws 6 engaged cog-wheels, and marks each of them with a clockwise or counter-clockwise arrow alternatively. "With 6 wheels, it all turns well, but if I put one more (he draws a small wheel between one pair of wheels and draws two arrows beside it - one clockwise and the other counter-clockwise - very close to the two wheels it is contacting); this wheel prevents the others from turning. It is an odd number. It is like with 5 wheels with respect to 4. If they are odd, they can not turn." Generally speaking, a PGC4 consists in a reasoning which can be described as follows: the regularity found in a particular generated case can put into action "expansive" research of a "general rule" whose particular starting case was an example; during research, new cases can be generated (cf. Pierce's "abduction" ; see Arzarello et al. 1998) 4.2. Some comments We may observe that PGC1 and PGC2 are, to some extent, dual processes. Indeed, in the first case mental exploration (centered on B) leads to detection of A as an arrival point, while in the second case the starting point is the regularity, and then dynamic exploration starts (by transforming the situation where the regularity occurs). We may wonder whether there is a common underlying cognitive background. N. Douek (personal communication) suggests that in PGC1, exploration leads to the "cause" that originates B, while in PGC2, exploration reveals the "cause", whose lack may make B fail to occur. So links emerge with "causality" as one of the possible backgrounds of conditionality (cf French, 1985).    PGC3 and PGC4 too are, to some extent, dual processes: in PGC3 extensive exploration leads to intensive insight; in PGC4 intensive exploration leads to a local insight, which in turn gives rise to extensive exploration that may confirm it and make it more exhaustive. N. Douek (personal communication) suggests that PGC3 implies the passage from the analytical description of several cases to an expression able to synthetize (some of) them while PGC4 involves the passage from a more or less synthetic expression of a particular case to a more general one suitable for wider application. In both cases, the passage from one representation to another seems to play a major role.    Bearing in mind preceding descriptions of PGCs and comments, we may expect that apriori analysis of the task (formulation and content) could to some extent predict the PGCs that will be produced by students. In particular, in a task aimed at discovering a singularity, we may expect that most PGCs will be of the PGC1 and PGC2 type, while in a "generalization" task most PGCs should be of the PGC3 and PGC4 types. The examined protocols confirm this prediction. For instance, in the case of the "generalizing and proving" task of EX. 3.1 and EX.4.1. only PGC3 and PGC4 were detected in the 43 protocols examined (with the exception of one student who produced his conjecture through a PGC2-type exploration).    In the examples considered before, students produce only one PGC; in general, we observed that in some cases the same student tries and abandons different PGCs before getting a conjecture he/she finds satisfactory. But in the case of the undergraduate students we also noticed quite frequently that a generation of conditionality can be reached through a sequence of coordinated steps, each of which bears a peculiar PGC (possibly different from those found in the other steps). 4.3. Some links between PGCs and construction of proof We have detected an important link between the PGCs described in the preceding subsection and students' proving processes under the same task: frequently the same mental exploration which leads to the conjecture is re-started by the student with entirely different functions during their proving process.    For example, as concerns PGC1, exploration can move from a support to the selection and the specification of the conjecture (in the conjecturing phase), to a support for the implementation of a logical connection (in the proving phase): some examples are reported in Garuti et al (1996). Here is reported the beginning of the proof produced by the student of EX. 1.1.: "I draw again the situation in a careful way. (she draws angle, bisecting line and the perpendicular passing by the chosen point, then another straight line passing through this point. She shadows the outcoming and incoming triangles) I see that the outcoming triangle is much bigger that the incoming triangle. But why is it bigger? Let us see. The area is base multiplied by height (she draws the heigths coming out from the vertex lying on the bisecting line). Yes, the heigths are equal and the bases are very different (etc.) " (The words in red put into evidence the point where the resumed exploration of the situation becomes explicitly functional to proving). As concerns PGC4: during the proving process, some students re-start from the particular case in which the regularity was detected, and then extend to other cases in order to find appropriate links between the hypotheses and the thesis; the function of the exploration changes from "what regularity this is a case of " to "what link this is a case of". An example is the proof by the student of EX. 4.1.: "We start again from 8 consecutive odd numbers; we try to write in that case the formula;: 2K+1+2K+3+2K+5+...+2K+15=(2K).8 + 1+3+5+...+15. I see that I obtain the sum of 8 consecutive odd numbers; also this sum must be divisible by 16; but this fact reminds me that in the sum of 15 numbers there is the factor 16… No, here we do not have 15 numbers; but we have all the odd numbers from 1 to 15. I have to find the formula that fits this case. I am remembering perhaps that it is n2 . Let us check. With 2 odds it is 4. With 4 odds it is 1+3+5+7=16. It is OK. The square of 4. But how can the square of 8 be divisible by 16? Yes, it is. 64=16x4. Let us check if it is true in general. n is even. Hence n2=(2m)2 is divisible by 2n". (The words in red put into evidence "abduction" phases during the proving process). Some students who had produced conditionality through a PGC3 also revealed this kind of behaviour in proving in the same task; and (on the contrary) some students who had produced conditionality through a PGC4 realized, during the proving process, an exploration similar to that exemplified in EX. 3.1. This seems to confirm the existence of deep links between PGC3 and PGC4 (as dual processes). 5. Possible Research Developments and Educational Implications The content of the preceding subsection raises interesting research problems about the links between the conjecturing process and the proving process in the perspective of the cognitive unity of theorems (see Garuti et al, 1998). In general, the exploration underlying a PGC and the exploration performed during the proving process are very similar in "nature" but differ in "function". What is the precise meaning of these two words? Another interesting research development concerns modeling of the possible links between PGCs and proof construction processes, especially when the task "Prove that.." requires the appropriation of a conjecture produced by others and then the production of new lemmas through PGCs, with related proofs (a typical situation in advanced mathematics).    And, naturally, the problem of identifying possible PGCs that differ from the four described in this paper still remains open.    Some connections with results produced by other researchers emerge from our analyses. We would particularly point out the need for in depth comparison of PGC2 and PGC4, and Balacheff's "crucial experiment" and "generic example", although the two criteria of analysis, (mainly "cognitive" in our case, and "epistemological" in the case of Balacheff) are different. Emerging connections bear deep, potential points of contact between epistemological and cognitive analyses.    As to the educational implications of this study, preceding analyses can be exploited to find appropriate tasks for students in different grades, so as to allow them to experience processes which seem to be relevant in mathematical activities concerning theorems. Indeed, we have remarked that the formulation and content of the task may influence students' PGCs (see 4.2.). Naturally, the interest lying in these considerations is related to an hypothesis of "educability" of the capacity to produce PGCs by experiencing them (cf. Simon, quotation in Section 2.). 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