Indirect proof: what is specific to this way of proving?
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- Antonini, S. & Mariotti, M.A. ZDM Mathematics Education (2008) 40: 401. doi:10.1007/s11858-008-0091-2
The study presented in this paper is part of a wide research project concerning indirect proofs. Starting from the notion of mathematical theorem as the unity of a statement, a proof and a theory, a structural analysis of indirect proofs has been carried out. Such analysis leads to the production of a model to be used in the observation, analysis and interpretation of cognitive and didactical issues related to indirect proofs and indirect argumentations. Through the analysis of exemplar protocols, the paper discusses cognitive processes, outlining cognitive and didactical aspects of students’ difficulties with this way of proving.
KeywordsProof Argumentation Indirect proof Proof by contradiction Proof by contraposition
Proving in an indirect way, by contradiction or by contraposition, is a common practice in the activity of mathematicians. Many of the most famous proofs are indirect: some proofs of the existence of infinite prime numbers, of the irrationality of the square root of 2, of the relationships between parallelism of two lines and the angles they form when intersected by a transversal, and many others. Although some of these proofs are already present in ancient mathematics, for example in Euclid’s Elements, and although some scholars (see Szabó 1978) support the thesis that proof by contradiction had a fundamental role in the origin of the concept of mathematical proof, indirect proof has frequently given rise to debates throughout the history of mathematics. In many cases, indirect proof has acquired a particular status among the arguments used in mathematics and, during such debates, some doubts on the acceptability of indirect proof as a mathematical proof have been discussed.
The debate raised by intuitionists at the beginning of the twentieth century is well known. Their refusal of both proof by contradiction and proof by contraposition was based on the refusal of the law of excluded middle and finally on a different interpretation of logical connectives (Dummett 1977, pp. 9–31).
Different positions were taken on the role of causality and, in particular, some mathematicians supported the elimination of proofs by contradiction (Mancosu 1996, pp. 24–28).
There was thus a consensus on the part of these scholars that proofs by contradiction were inferior to direct proofs, on account of their lack of causality. The consequences to be drawn from this position are of relevance to the foundations of classical mathematics. (Mancosu 1996, p. 26)
Generally speaking, mathematicians recognize that proof by contradiction may ask for a mental effort. It may be too demanding to assume that what is to be proved is false, and it is extremely hard for one’s mind to follow the deductive steps when false hypotheses and contradictions are involved. Our study aims to investigate the nature of this effort, outlining cognitive and didactical aspects of students’ difficulties with indirect proof.
To find a not obvious proof is a considerable intellectual achievement but to learn such a proof, or even to understand it thoroughly costs also a certain amount of mental effort. Naturally enough, we wish to retain some benefit from our effort, and, of course, what we retain in our memory should be true and correct and not false or absurd. (Polya 1945, p. 168)
2 Indirect proof
First of all, let us clarify what we mean with the expression indirect proof. The use of the expressions ‘indirect proof’, ‘proof by contradiction’, ‘proof by contraposition’, ‘proof ad absurdum’ in the textbooks is far from being clear and uniform, and it may be considered controversial even among the mathematicians (Antonini 2003a; Bernardi 2002). In particular, if a statement S can be expressed as an implication p→q, a proof by contraposition of S is a direct proof of ¬q→¬p, while a proof by contradiction of S is a direct proof of pΛ¬q→rΛ¬r where r is any proposition. In Italy, in general, mathematicians and teachers call ‘proof by contradiction’ (Italian: ‘dimostrazione per assurdo’) both proof by contraposition and proof by contradiction. Bellissima and Pagli (1993) explain this fact by saying that what seems to be psychologically meaningful in proof by contradiction is the starting point that is the negation of the thesis. This characteristic is also shared by proof by contraposition. In spite of significant differences, we can point out some important commonalities of these types of proof. Therefore, in this paper, we deal with both proof by contradiction and proof by contraposition, referring to them through the term indirect proof.
From a cognitive and didactical point of view, there are not many studies in which indirect proof is studied. Nevertheless, even if these studies are enacted from different points of view, they report that, at any school level, students’ difficulties with indirect proof seem to be greater than those related to direct proof.
Different interpretations and different sources of these difficulties were proposed. Some authors remarked that indirect proofs are not given adequate attention in school practice, at any school level (Bernardi 2002; Thompson 1996). This educational reason cannot fully explain the difficulties that students seem to face. Some studies contributed to identify some particular aspects of indirect proof and of students’ cognitive processes that might give insight into roots of their difficulties.
Hence, no construction of the results of the theorem is enacted. Indeed, at the end of the proof, as soon as a contradiction is deduced, the ‘false world’ has to be rejected, and students can feel confused and dissatisfied because of the unexpected destruction of the mathematical objects on which the proof was based.
In indirect proofs, however, something strange happens to the ‘reality’ of these objects. We begin the proof with a declaration that we are about to enter a false, impossible world, and all our subsequent efforts are directed towards ‘destroying’ this world, proving it is indeed false and impossible. We are thus involved in an act of mathematical destruction, not construction. Formally, we must be satisfied that the contradiction has indeed established the truth of the theorem (having falsified its negation), but psychologically, many questions remain unanswered. What have we really proved in the end? What about the beautiful constructions we built while living for a while in this false world? Are we to discard them completely? And what about the mental reality we have temporarily created? I think this is one source of frustration, of the feeling that we have been cheated, that nothing has been really proved, that it is merely some sort of a trick—a sorcery—that has been played on us. (Leron 1985, p. 323)
In the case of proof by contraposition, many authors underline the problem of the students’ acceptability of the proof. Fischbein (1987, pp. 72–81) claims that the modus tollens, the inference rule that justifies the method of proof by contraposition and proof by contradiction, is not as intuitive as the inference rule of modus ponens. Stylianides, Stylianides and Philippou (2004) describe how verbal and symbolic aspects may affect students’ performances when dealing with the equivalence between a statement and its contrapositive.
The history of mathematics shows how the role assigned to proof was sometimes at the origin of problems concerning the acceptability of indirect proof. Starting from the distinction between different functions of a proof, Barbin (1988) focuses on the explanatory function and interprets students’ difficulties in accepting indirect proof, on the consideration that this method of proof does not lead to insight, and, in particular, to the discovery of new statements.
Thus, beyond the analysis of the difficulties in understanding and producing indirect proofs, it seems reasonable to enlarge our discussion to an epistemological and cognitive analysis including the conjecturing process which leads to the production of a new statement. In particular, we discuss the complex relationship between arguments supporting a statement and its validation by a mathematical proof.
3 Argumentation and proof
Epistemological and historical analyses led some authors (Duval 1995, 1992–1993) to claim a distance and sometimes even a cognitive rupture between argumentation and mathematical proof. As the author explains, argumentation may be regarded as a process in which the discourse is developed with the specific aim of making an interlocutor change the epistemic value given to a particular statement. In short, argumentation consists of whatever rhetoric means are employed in order to convince somebody of the truth or the falsehood of a particular statement. On the contrary, proof consists of a logical sequence of implications that states the theoretical validity of a statement.
Indirect argumentation seems to be a natural way of thinking, and as some authors report, students spontaneously produce argumentation with indirect structure, also in mathematics. They do that in order to generate conjectures, to convince themselves or others of the truth of some statements, or to understand why a statement is true (Antonini 2003b; Reid & Dobbin 1998; Thompson 1996; Freudenthal 1973; Polya 1945). Didactical implications related to this data have been suggested. For instance, Thompson writes:
The indirect proof is a very common activity (‘Peter is at home since otherwise the door would not be locked’). A child who is left to himself with a problem, starts to reason spontaneously ‘...if it were not so, it would happen that...’ (Freudenthal 1973, p. 629)
It becomes necessary, in addition to being interesting, to investigate both indirect argumentation and indirect proof, as they are produced by students; we hypothesize that continuity and ruptures can occur, and we are interested in making explicit some aspects that characterize these possibilities. The following section is devoted to the introduction of a specific theoretical framework, within which our investigation can be developed: focussing on similarities without neglecting the differences, we model the relationship between argumentation and proof by the notion of Theorem and that of Cognitive Unity.
If such indirect proofs are encouraged and handled informally, then when students study the topic more formally, teachers will be in a position to develop links between this informal language and the more formal indirect-proof structure. (Thompson 1996, p. 480)
3.1 The notions of Theorem and Cognitive Unity
Investigations on the relationship between mathematical proofs and the process of argumentation produced interesting results on a possible continuity rather than a rupture between them, and led to the elaboration of the theoretical construct of Cognitive Unity. The term ‘Cognitive Unity’ was initially coined to express a hypothesis of continuity in the context of the solution of open-ended problems (Garuti, Boero & Lemut 1998), and it was later redefined (Pedemonte 2002) to express the possibility of congruence between some aspects of the argumentation phase and the subsequent proof produced. In this re-elaboration, it was clearly assumed that such congruence may or may not occur.
The existence of a reference theory as a system of shared principles and deduction rules is needed if we are to speak of proof in a mathematical sense. Principles and deduction rules are so intimately interrelated so that what characterises a mathematical theorem is the system of statement, proof and theory. (Mariotti, Bartolini Bussi, Boero, Ferri & Garuti 1997, pp. 182–183)
The construct of Cognitive Unity provides a perspective from which we observe the relationship between argumentation and proof by focussing on analogies, without forgetting the differences. Cognitive Unity offered a great potential in framing our investigation. Moreover, it allows taking into account both epistemological and cognitive considerations and it sheds light onto the complex relationship between the individual and the cultural dimensions of mathematics.
The analysis of argumentations and proofs according to both these constructs may reveal analogies as well as discrepancies. In particular, different structures of mathematical proofs can be compared with the structures produced by the analysis of observable argumentations (Pedemonte 2007). What is interesting is the fact that not only some of the observable argumentations present a structure that is not mathematically acceptable, but also that some of the mathematically acceptable logical structures are not as acceptable as one could expect. This seems to be the case, in particular, for some occurrences of indirect proof.
4 Towards an interpretative model
The elaboration of the model of theorem with an indirect proof is based on the ‘didactic’ notion of mathematical theorem, as introduced above. According to such characterization, a mathematical theorem consists in the system of relations between a statement, its proof, and the theory within which the proof makes sense. In this paper, we will refer to the triplet constituted by statement, proof and theory as (S, P, T).
We notice that in the triplet (S, P, T) there are no limitations on the type of proof (direct, indirect, by induction, etc.). Moreover, the third component, the theory T, stands for both the mathematical theory—as Euclidean Geometry, Number Theory, and so on—and the logical theory of inference rules. The refinement of this triplet was elaborated with the aim of taking into account the basic aspects of indirect proof, that are its logical structure and the distinction between theory and meta-theory.
Let us consider two examples in which a proof by contraposition and a proof by contradiction are provided.
4.1 Example 1
Statement: Let n be a natural number. If n 2 is even then n is even.
Proof: Assume n to be a natural odd number, then there exists a natural number k such that n = 2k + 1. As a consequence n 2 = (2k + 1)2 = 4k 2 + 4k + 1 = 2(2k 2 + 2k) + 1, then n 2 is an odd number.
This is an example of proof by contraposition. The given proof is a direct proof of the statement “if n is odd then n 2 is odd”, that is the contraposition (¬q→¬p) of the original statement (p→q).
4.2 Example 2
Statement: Let a and b be two real numbers. If ab = 0 then a = 0 or b = 0.
Proof: Assume that ab = 0, a ≠ 0, and b ≠ 0. Since a ≠ 0 and b ≠ 0 one can divide both sides of the equality ab = 0 by a and by b, obtaining 1 = 0.
This is an example of a proof by contradiction, where a direct proof of the statement “let a and b be two real numbers; if ab = 0, a ≠ 0, and b ≠ 0 then 1 = 0 ” is given. The hypothesis of this new statement is the negation of the original statement and the thesis is a false proposition (“1 = 0”).
Principal statement and secondary statement involved in two indirect proofs
Principal statement S
Secondary statement S*
Let n be a natural number
If n 2 is even then n is even
Let n be a natural number
If n is odd then n 2 is odd
Let a and b be two real numbers
If ab = 0 then a = 0 or b = 0
Let a and b be two real numbers
If ab = 0, a≠0, and b≠0 then 1 = 0
Therefore, in both proof by contraposition and proof by contradiction we can identify the shift from one statement (principal statement) to another (secondary statement).
From the point of view of logic, we have to justify the acceptability of the proof of the secondary statement (S*) as a proof of the principal statement (S). In particular, this requires the validity of the statement S*→S. Moreover, in this case it is possible to derive the validity of S from S* and S*→S by the well-known modus ponens inference rule. But the validity of the implication S*→S depends on the logical theory, within which the assumed inference rules are stated. As commonly occurs, i.e. in the classic logical theory, such a theorem is valid, but this is not the case in other logical theories, such as the minimal or the intuitionistic logic (see Prawitz 1971).
Therefore, it is necessary to have a theorem in order to validate the principal statement. This theorem is not part of the theory in which the principal and secondary statements are formulated, but it is part of the logical theory. Referring to their meta-theoretical status, we call the statement S*→S meta-statement, the proof of S*→S meta-proof, and the logical theory, in which the meta-proof makes sense, meta-theory.
4.3 A model of indirect proof
the sub-theorem (S*, C, T) consisting of the statement S* and a direct proof C based on a specific mathematical theory T (Algebra, Euclidean Geometry, and the like);
a meta-theorem (MS, MP, MT), consisting of a meta-statement MS = S*→S and a meta-proof MP based on a specific meta-theory MT (that usually coincides with classic logic);
the principal theorem, consisting of the statement S and the indirect proof of S, based on a theoretical system consisting of both the theory T and the meta-theory MT.
5 Difficulties with indirect proof
The model we have set up is useful to analyze the structure of indirect proof by identifying some elements that are specific to this type of proof, and that we hypothesize could be critical for students. In the following sections, we use the model both to analyze indirect proof and to describe (and analyze) students’ cognitive processes, involved both in producing and interpreting indirect proofs.
5.1 The theory of reference in the sub-theorem of a proof by contradiction
When mathematicians prove a statement, they call it a “true” statement. Such “truth” is in relation to a specific semantic of the theory within which the proof is provided. Moreover, the truth of a valid statement is drawn from accepting both the hypothetical truth of the stated axioms and the fact that the stated rules of inference “transform truth into truth”.
Statement S*: although the hypothesis and the thesis are false, S* makes sense from a logical point of view. Moreover, since its hypothesis is false, according to the truth tables, the implication S* results to be true.
Proof C: it constitutes a valid proof of the implication S*. It means that it is possible to construct a deductive chain within a mathematical theory, and this, despite the fact that both the hypothesis and the thesis are false. That means something more than the fact that S* is true according to the truth tables.
Theory T: deduction in a theory is independent from the interpretation of the statements involved. That means that axioms and theorems of a theory can be applied to objects which are ‘impossible’. For instance, it is possible to apply the given theory to two real numbers a and b different from 0 and such that ab = 0, to the rational square root of 2, to parallel lines that intersect reciprocally, and so on.
(Principal) statement: let a and b be two real numbers. If ab = 0 then either a = 0 or b = 0.
Proof: assume that ab = 0, a ≠ 0, and b ≠ 0. Since a ≠ 0 and b ≠ 0, both sides of the equality ab = 0 can be divided by a and by b, obtaining 1 = 0.
This is a direct proof of the secondary statement “let a and b be two real numbers; if ab = 0, a ≠ 0 and b ≠ 0, then 1 = 0”. The hypothesis of this statement is “there exist two real numbers a and b such that ab = 0, a ≠ 0 and b ≠ 0”. Such hypothesis is false because—according to a standard semantics—there are no real numbers a and b such that ab = 0, a ≠ 0 and b ≠ 0. The thesis is “1 = 0”, it is false because 1 ≠ 0. On the contrary, the implication expressed by the statement is true, according to the truth tables and because its hypothesis is false.
for any real number x, if x ≠ 0, then there exists a number y such that xy = 1;
for any real numbers x, y, z, if x = y then xz = yz.
In summary, the validity of the proof of the secondary statement is based on the validity of a deductive chain within the theory T that is applied to ‘impossible’ objects.
5.2 Difficulties in identifying the theory of reference
From a cognitive point of view, the peculiarity of the sub-theorem may have serious consequences. In particular, conflicts may arise between the theoretical and the cognitive points of view.
Some authors, for instance Durand-Guerrier (2003), pointed out some students’ difficulties in evaluating or accepting the truth-value of an implication having a false antecedent. In the following, we are interested in showing students’ difficulties in evaluating both the truth-value and, in particular, the validity and the acceptability of a deductive chain starting from false assumptions and requiring to manage the mathematical theory of reference applied to ‘impossible’ objects (see also Mariotti & Antonini 2006).
I: Could you try to prove by contradiction the following: “if ab = 0 then a = 0 or b = 0”?
M: [...] well, assume that ab = 0 with a different from 0 and b different from 0... I can divide by b... ab/b = 0/b... that is a = 0. I do not know whether this is a proof, because there might be many things that I haven’t seen.
M: Moreover, so as ab = 0 with a different from 0 and b different from 0, that is against my common beliefs [Italian: “contro le mie normali vedute”] and I must pretend to be true, I do not know if I can consider that 0/b = 0. I mean, I do not know what is true and what I pretend it is true.
I: Let us say that one can use that 0/b = 0.
M: It comes that a = 0 and consequently … we are back to reality. Then it is proved because … also in the absurd world it may come a true thing: thus I cannot stay in the absurd world. The absurd world has its own rules, which are absurd, and if one does not respect them, comes back.
I: Who does come back?
M: It is as if a, b and ab move from the real world to the absurd world, but the rules do not function on them, consequently they have to come back …
M: But my problem is to understand which are the rules in the absurd world, are they the rules of the absurd world or those of the real world? This is the reason why I have problems to know if 0/b = 0, I do not know whether it is true in the absurd world. […]
I: [The interviewer shows the proof by contradiction of the statement “ \( \sqrt 2 \) is irrational”, then asks:] what do you think about it?
M: in this case, I have no doubts, but why is it so? … perhaps, when I have accepted that the square root of 2 is a fraction I continued to stay in my world, I made the calculations as I usually do, I did not put myself problems like ‘in this world, a prime number is no more a prime number’ or ‘a number is no more represented by the product of prime numbers’. The difference between this case and the case of the zero-product is in the fact that this is obvious whilst I can believe that the square root of 2 is a fraction, I can believe that it is true and I can go on as if it were true. In the case of the zero-product, I cannot pretend that it is true, I cannot tell myself such a lie and believe it too!
According to our model, Maria’s difficulties concern the sub-theorem (S*, C, T) and, in particular, the identification of the theory T, to which the proof C refers. Maria claims that where we accept something as false, anything can happen, including 0/b ≠ 0. The absurdity of the hypothesis of the secondary statement is in conflict with the use of the ‘common’ theory, and Maria thinks that such theory T should be replaced by a new theory T* (8), that might be more adequate with respect to the “absurd world” generated by the false assumption and in which the proof makes sense.
The case of irrationality of \( \sqrt 2 \) is different. In this proof, Maria does not feel that an ‘absurd world’ is involved, because the fact that \( \sqrt 2 \) is rational is acceptable for her (10). Consequently, the basic truths are not questioned (“I can believe that it is true and I can go on as if it were true”) and the theory of reference is not disturbed.
5.3 Difficulties in the shift from the principal to the secondary statement
Statements, proofs and theoretical levels involved in a theorem with indirect proof
(S*, C, T) + (MS, MP, MT) indirect
T + MT
Theory and meta-theory
From the point of view of logic, the role of the meta-theorem is fundamental: the meta-statement S*→S is a statement that can be proved only within some meta-theories, but not within others. As mentioned above, if the statement S*→S cannot be proved there is the remarkable consequence that proof by contradiction and proof by contraposition are not valid modes of inference.
From the point of view of our model, some of the difficulties highlighted in the literature (Stylianides, Stylianides & Philippou 2004; Antonini 2004; Fischbein 1987, pp. 72–81) can be described and interpreted in terms of the complexity that the move from the theoretical level to the meta-theoretical level requires (see also Antonini & Mariotti 2007).
F: Proof by contradiction is artificial: how does one get out of it? Ok, you have arrived to the contradiction… and then? […] I don’t see that conclusion be linked to the other one, I miss the spark […]
I: Let’s think of an example: we take a natural number n. Theorem: if n2 is even then n is even. Proof: if n is odd I write n = 2k + 1, then... [the interviewer writes down algebraic transformations] n 2 = 2(2k 2 + 2k) + 1 is odd.
F: Yes, I understand, it is better to prove that if n is odd then n 2 is odd.
I: And then, what is the problem?
F: The problem is that in this way we proved that n is odd implies n 2 is odd, and I accept this; but I do not feel satisfied with the other one.
I: Do you agree that natural numbers are odd or even and there are not other possibilities?
F: Yes, of course... and now you will say: n 2 is even, n is even or odd, but if it were odd, n 2 would be odd, but it was even... yes, ok, I know, but… I’m not getting something.
F: First of all, why do I have to begin from n not even? I don’t see any immediate conclusion. And, at the end: ‘then it cannot be other than n even’, it is a gap, the gap of the conclusion... it’s an act of faith... yes, at the end it’s an act of faith.
F: Yes, there are two gaps, an initial gap and a final gap. Neither does the initial gap is comfortable: why do I have to start from something that is not? […] However, the final gap is the worst, […] it is a logical gap, an act of faith that I must do, a sacrifice I make. The gaps, the sacrifices, if they are small I can do them, when they all add up they are too big.
F: my whole argument converges towards the sacrifice of the logical jump of exclusion, absurdity or exclusion… what is not, not the direct thing. Everything is fine, but when I have to link back… [Italian: “Tutto il mio discorso converge verso il sacrificio del salto logico dell’esclusione, assurdo o esclusione… ciò che non è, non la cosa diretta. Va tutto bene, ma quando mi devo ricollegare...”]
Subsequently (2), the interviewer proposes a theorem with a proof by contraposition. The principal statement is
S : if n 2 is even then n is even.
The proof consists of the direct proof of the secondary statement, that remains unspoken,
S*: if n is odd then n 2 is odd.
Fabio makes explicit that what it is proved is the secondary statement S* (3). Moreover, it is relevant that Fabio is aware (3) that “it is better” (easier?) to prove S*. Nevertheless, Fabio clearly expresses his feelings: he can identify the two statements (5), he accepts the given proof as a proof of S* (“I accept this”) but not as a proof of S (“I do not feel satisfied with the other one”).
The method of indirect proof seems clear to Fabio who is able to produce an argument to explain it (7). Nevertheless, there is something that he is not able to grasp (“I’m not getting something”). The shift from the proof of the secondary statement to the validation of the principal statement is not immediate, is not rationally acceptable.
What makes this protocol so peculiar is the fact that Fabio’s ability of introspection lets us know where the conflict arises. In fact, Fabio openly expresses his feeling of distress.
According to our model, the difficulty can be localized in the cognitive difficulty of grasping as immediate and intuitive the logical link expressed by the meta-statement S*→S. For Fabio, and probably for many other students, such a link is not immediate (we could say ‘an intuition’ in the sense of Fischbein 1987) and its acceptance causes distress (“I do not feel satisfied”, “I’m not getting something“, “I don’t see any immediate conclusion”, “everything is fine, but when I have to link back…”, etc.).
It is also interesting to notice the metaphors used to described the shift between the two statements and the feeling he faces. Fabio talks about “gaps” and about something that he has to “link”. Moreover, with the word “sacrifice” he expresses, in a very dramatic way, the cognitive effort he has to do in order to “link back” to what is detached.
6 Indirect argumentation
In the previous sections, through the analysis of different elements of indirect proof and of their relations, we showed some of the difficulties that students could meet when engaged in indirect proof.
Certainly, the complexity of the logical structure of indirect proof, as highlighted by the model, can explain the difficulties met by the students, but from the perspective of Cognitive Unity, it is reasonable to put forth the question whether similar difficulties can be found in the production of indirect argumentations.
As previously mentioned, results coming from the recent literature and from our own experiments show that students spontaneously produce indirect argumentations. Therefore, we are interested in investigating what makes indirect argumentation spontaneously acceptable. In particular, we are interested in studying those aspects that allow one to overcome the obstacles and difficulties that were highlighted.
In the following, we analyze some indirect argumentations that students spontaneously produced when they asked to generate a conjecture. We will see how the model is useful in identifying, describing and analyzing indirect argumentations, and comparing them to indirect proofs.
6.1 Indirect argumentation and meta-theorem
The subjects involved in the interview are two secondary school students, Valerio (grade 13) and Cristina (grade 11),1 who have had a lot of experience in the field of Euclidean Geometry.
The proposed task is an open-ended problem in geometry (see Antonini 2003b):
two lines r and s lie on a plane, and have the following property: each line t intersecting r, intersects s, too. Is there anything you can say about the reciprocal position of r and s? Why?
To simplify the exposition, we call A the property ‘each line t intersecting r, intersects s, too’. With this notation, the problem is of the form ‘given A, what can you deduce?’
After an exploration phase, during which students try to make sense of property A, Valerio proposes some conjectures supported by an argumentation.
31 V: Well, it [line t] cannot be parallel to any of the two lines because, if we have two crossing lines, even if they are not perpendicular, if it [line t] is parallel to one of the two, it intersects only one of them.
32 C: Yes, it’s the same situation of the two perpendicular lines.
33 I: Then?
34 V: We had to discuss the reciprocal position of r and s.
35 C: They cannot be either crossing lines or…
36 V: They cannot be crossing lines.
37 C: Yes. If they are perpendicular we know …
38 V: Perpendicular…
39 C: Er, if they are parallel then we have …
40 V: Oh yes, then they [r and s] definitely have to be parallel.
41 C: Parallel.
42 I: Why?
43 V: Because, they will never intersect each other if they are parallel.
44 C: Because…
45 V: They will never intersect each other and then there cannot be a situation like this [he points at his drawing, see Fig. 1], in which, since they [r and s] cross, the line t is parallel to r or to s and then it [t] does not intersect both.
46 C: The line…
47 V: […] If they [r and s] are not parallel there will be always a point in which they intersect, there can always be a situation in which there is a line parallel to only one of them, which then intersects only one line.
S1: (If A is true then) r and s are not perpendicular (21)
S2: (If A is true then) r and s are not crossing lines (31)
S3: (If A is true then) r and s are parallel lines (40)
We think that the negative form in which the statements are formulated (r and s are not …) makes immediate the students’ transition from the secondary statement (if r and s are perpendicular/crossing then A is not true) to the principal statement (if A is true, r and s are not perpendicular/crossing).
The case of the argumentation supporting S3 is different. Although S2 and S3 are logically equivalent, in S3 the negation disappears. Transition from S2 to S3 does not take much time but it is far from being immediate. It requires a collaborative work of elaboration and successive reformulations (33–40), and this process seems fundamental. Different cases are considered (perpendicular, intersecting, parallel lines) showing the students’ worry of not neglecting any case. The formulation of S3 is supported by the explicit remark about the fact that the case of parallelism excludes all the others, as Valerio explains in response to the request of the interviewer (43). The list of cases is still the grounding of the first argumentation of S3 that Valerio proposes (45). Such argumentation is indirect and logically incorrect. On the contrary, the final argumentation (47) is correct and seems to condense the whole process. The production of arguments in the different cases seems to be a necessary prerequisite, before those arguments can be condensed in the hypothesis of the secondary statement S3* (“r and s are not parallel”) and in the S3* supporting argument (47). In other terms, we assume that this process of elaboration played the role of a meta-argument, corresponding to the role played by the meta-theorem in our model. Similarly to what happens for the meta-theorem, such a meta-argument supports the validity of the indirect argumentation, allowing the students to bridge the gaps between the secondary statement (‘if r and s are not parallel lines then A is false’) and the principal statement (‘if A is true then r and s are parallel’). In fact, after this claim the students stop and seem to be satisfied.
6.2 Indirect argumentation and reference theory
The aim of the following example is to show the spontaneous production of an indirect argumentation supporting a conjecture, and some difficulties arising in the construction of the proof of the conjectured statement. The analysis of the protocol, carried out in the frame of our model, highlights some difficulties in the application of the theory of Euclidean Geometry to an object that is geometrically inconsistent and how these difficulties can be overcome in argumentative processes.
The two students, Paolo and Riccardo (grade 13), are high achievers, according to the evaluation provided by their teachers. The open-ended problem proposed is the following:
What can you say about the angle formed by two bisectors in a triangle?
61 P: As far as 90°, it would be necessary that both K and H are 90°, then K/2 = 45, H/2 = 45...180°−90° and 90°.
62 I: In fact, it is sufficient that the sum is 90°, that K/2 + H/2 is 90°.
63 R: Yes, but it cannot be.
64 P: Yes, but it would mean that K + H is ... a square […]
65 R: It surely should be a square, or a parallelogram
66 P: (K − H)/2 would mean that […] K + H is 180°...
67 R: It would be impossible. Exactly, I would have with these two angles already 180°, that surely it is not a triangle.
71 R: We can exclude that [the angle] is π/2 [right] because it would become a quadrilateral.
That argument is based on theoretical considerations, precisely on the theorem about the sum of the angles of a triangle. The theory is applied to a virtual geometrical figure, that at the beginning is a triangle and at the end is a quadrilateral: in other words, the figure is modified in order to respect the relationships expressed by the theory. The quadrilateral seems to emerge from reasoning based on a dynamic mental image elaborated within the current Geometrical Theory. The deformation of the original triangle into a quadrilateral, as a consequence of the construction of a right angle, can be considered a compromise between the new hypotheses and the available theory. In other words, it can be interpreted as an antidote for an ‘absurd world’.
A meaningful difference between this argumentation and a mathematical (indirect) proof is in the application of the theory to the geometrical figure. We think that this difference is the main source of the difficulties that the students faced in constructing an indirect proof. In Riccardo’s argumentation, the theory is applied to a geometrical figure that is changed according to the validity of the theorems he knows. In the mathematical proof, the theory applied to the impossible geometrical figure leads to a contradiction: the geometrical figure is not modified but refused by means of the meta-theorem.
Note that the argumentation is accepted even if there is nothing in Riccardo’s argumentation that is explicitly referred to the meta-theorem. Once again, as in the case of Valerio and Cristina, a fundamental role is played by the consideration of different possibilities: the figure can be a triangle, a square, or a parallelogram. The fact that the angle S is a right angle is not excluded because of a contradiction. Instead, it is excluded by the determination of a well-defined figure, as the consequence of the angle S being right. This final figure is a quadrilateral and this excludes the case of the triangle. The arguments, by which it was possible to determine a figure and to show that it is not a triangle, are very convincing, and perhaps stronger than any argument based on a contradiction. This may explain the immediate acceptability of this indirect argumentation.
We proposed a model through which to analyze proof and argumentation having indirect structure. By analyzing specific aspects of indirect proof, the model revealed its efficiency in identifying, analyzing and interpreting students’ difficulties when dealing with this method of proof.
Moreover, the analysis of students’ argumentations in open-ended problems highlights, on one hand, some important differences between indirect argumentation and indirect proof, and, on the other hand, how some difficulties can be overcome. For example, the protocol of Riccardo and Paolo reveals a meaningful different treatment of the reference theory in argumentation and proof. In argumentation, the need of preserving the theory of reference can lead one to transform the geometric figure on which the argument is focused. On the contrary, in an indirect proof, the application of the theory in the deductive chain results in a contradiction. Moreover, the protocols we presented, in particular that of Valerio and Cristina, show how the students can bridge the gap between the principal statement and the secondary statement by producing an argument in which different cases are classified.
The previous discussion suggests that the Cognitive Unity approach can also be an efficient didactical tool for designing teaching/learning situations aimed to introduce indirect proofs.
For this student, some functions of proof seems not to be present in a proof by contradiction: this type of proof does not convince him because it is not a method to generate a conjecture (“I have to know in advance what I have to prove”), and it is not an argumentation to support the statement (“I have to be convinced in some way that what I have to prove is true”).
“Proofs by contradiction do not convince me, because I have to know in advance what I have to prove, while with direct proof I can rearrange the arguments, modify the direction during the proof [Italian: “correggere il tiro strada facendo”] [...] To use proofs by contradiction I have to be convinced in some way that what I have to prove is true.”
Yet, we think that the Cognitive Unity approach can go further. As we showed, the production of indirect argumentation can hide some cognitive processes, whose roles are very significant in the production and the acceptability of indirect proof. The activity of producing a conjecture can offer students the possibility of activating these processes and then of constructing a bridge to overcome the gaps that indirect proof seems to provoke. On the contrary, without any conjecturing phase, some gaps could not be bridged or could require sacrifices and mental efforts that not all the students seem to be able to make.
Valerio and Cristina do not belong to the same class, although they belong to the same school.