ORIGINAL ARTICLE

Indirect proof: what is specific to this way of proving?

Samuele Antonini Æ Maria Alessandra Mariotti

Accepted: 14 April 2008 / Published online: 7 May 2008

� FIZ Karlsruhe 2008

Abstract 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 state-

ment, a proof and a theory, a structural analysis of indirect

proofs has been carried out. Such analysis leads to the pro-

duction 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.

Keywords Proof � Argumentation � Indirect proof �Proof by contradiction � Proof by contraposition

1 Introduction

Proving in an indirect way, by contradiction or by con-

traposition, 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 Szabo 1978)

support the thesis that proof by contradiction had a fun-

damental 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).

Another paradigmatic example is the discussion,

developed in the sixteenth and seventeenth centuries.

Starting from the Aristotelian position that causality should

be the base of the scientific knowledge (Mancosu 1996),

the debate involved the acceptability of proofs (for a dis-

cussion related to education, see Harel 2007). In particular,

to be part of a scientific endeavour, a proof should proceed

from the cause to the effect. Therefore, according to the

Aristotelian point of view, a proof by contradiction could

not to reveal the cause since it is not based on true

premises. As Mancosu states:

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.

This research study was supported by the Italian Ministry of

Education and Research (MIUR) Prin 2005 # 2005019721.

S. Antonini (&)

Dipartimento di Matematica, Universita di Pavia,

Via Ferrata, 1, 27100 Pavia, Italy

e-mail: samuele.antonini@unipv.it

M. A. Mariotti

Dipartimento di Scienze Matematiche e Informatiche,

Universita di Siena, Piano dei Mantellini, 44,

53100 Siena, Italy

e-mail: mariotti.ale@unisi.it

123

ZDM Mathematics Education (2008) 40:401–412

DOI 10.1007/s11858-008-0091-2

The consequences to be drawn from this position are

of relevance to the foundations of classical mathe-

matics. (Mancosu 1996, p. 26)

Different positions were taken on the role of causality and,

in particular, some mathematicians supported the elimina-

tion of proofs by contradiction (Mancosu 1996, pp. 24–28).

Although nowadays the acceptability of this way of

proving is no longer an issue, mathematicians share the

opinion that proof by contradiction is peculiar. It is

remarkable what Polya writes:

To find a not obvious proof is a considerable intel-

lectual 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)

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.

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 pK:q?rK: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 con-

traposition. 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 con-

tradiction 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. Nev-

ertheless, even if these studies are enacted from different

points of view, they report that, at any school level, stu-

dents’ 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 con-

tributed to identify some particular aspects of indirect proof

and of students’ cognitive processes that might give insight

into roots of their difficulties.

Some difficulties seem to be at the beginning of the

indirect proof, related to the formulation and interpretation

of the negation of the thesis (Wu Yu, Lin & Lee 2003;

Antonini 2001, 2003a; Thompson 1996). In the case of

proof by contradiction, Leron (1985) identifies one of the

main difficulties in assuming and dealing with false

hypotheses:

In indirect proofs, however, something strange hap-

pens 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 psycho-

logically, 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)

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 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

402 S. Antonini, M. A. Mariotti

123

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 con-

cerning 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 under-

standing and producing indirect proofs, it seems reasonable

to enlarge our discussion to an epistemological and cog-

nitive 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 argumen-

tation 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 theoreti-

cal validity of a statement.

Difficulties faced by students in dealing with proof can

be related to the problematic relationship between the

theoretical status of formal proof and the cognitive and

pragmatic status of argumentation. As far as indirect proofs

are concerned, such a distance becomes more significant

and may explain some of the data reported in the current

literature. Inspite of the unanimously recognized difficul-

ties with indirect proofs, people spontaneously produce

argumentations where, although contradictions can have

different forms and functions (Balacheff 1991; Piaget

1974), an indirect structure is recognizable, as pointed out

by Freudenthal:

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 prob-

lem, starts to reason spontaneously ‘...if it were not

so, it would happen that...’ (Freudenthal 1973, p. 629)

Indirect argumentation seems to be a natural way of

thinking, and as some authors report, students spontane-

ously 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:

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)

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.

3.1 The notions of Theorem and Cognitive Unity

Proof is traditionally considered in itself, but it is not

possible to grasp the sense of a mathematical proof without

linking it to the other two elements: a statement, that the

proof provides a support and a theory, i.e. the theoretical

frame within which this support makes sense. With the aim

of expressing the complexity of this relation, the following

characterization of Mathematical Theorem was introduced:

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)

Indirect proof: what is specific to this way of proving? 403

123

Investigations on the relationship between mathematical

proofs and the process of argumentation produced inter-

esting 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 (Pede-

monte 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 construct of Cognitive Unity provides a perspective

from which we observe the relationship between argu-

mentation 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 rela-

tionship 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 dis-

crepancies. 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 struc-

tures 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 charac-

terization, 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 lim-

itations 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 n2 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

n2 = (2k + 1)2 = 4k2 + 4k + 1 = 2(2k2 + 2k) + 1, then

n2 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 n2

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 proposi-

tion (‘‘1 = 0’’).

In both examples, in order to prove a statement S, that

we will call the principal statement, a direct proof of

another statement S* is given. We will call S* the sec-

ondary statement (see Table 1).

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

Table 1 Principal statement and secondary statement involved in

two indirect proofs

Principal statement S Secondary statement S*

Let n be a natural number

If n2 is even then n is even

Let n be a natural number

If n is odd then n2 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

404 S. Antonini, M. A. Mariotti

123

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

According to the previous analysis, in any theorem with

indirect proof we can recognize two theoretical levels,

three statements, and three theorems:

(1) 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 Geome-

try, and the like);

(2) 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);

(3) 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.

We call indirect proof of S the pair consisting of the sub-

theorem (S*, C, T) and the meta-theorem (MS, MP, MT);

in symbols P = [(S*, C, T), (MS, MP, MT)]. In summary,

an indirect proof consists of a couple of theorems

belonging to two different logical levels: the level of the

mathematical theory and the level of the logical theory.

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’’.

In this paragraph, we analyze the sub-theorem defined

by the triplet (S*, C, T), where S* is a secondary statement,

C is a direct proof of S* and T is the mathematical theory

within which this proof is constructed and validated. In

particular, we refer to the specific case of proof by con-

tradiction. In this case, one of the main characteristics of

the theorem (S*, C, T) concerns the fact that both the

hypothesis and the thesis of the statement S* are false

propositions in a standard semantic. Still in respect to such

semantic, the following holds:

• 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.

For example, let us consider the previous theorem and

analyze its proof according to our discussion.

(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’’,

Indirect proof: what is specific to this way of proving? 405

123

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.

The reference theory is the theory of real numbers (or

more generally, Field Theory). In particular, the proof is

based on the following two axioms:

(1) for any real number x, if x = 0, then there exists a

number y such that xy = 1;

(2) for any real numbers x, y, z, if x = y then xz = yz.

These two axioms are applied to ‘impossible’ mathematical

objects: the axiom 1 is applied to the non-existing real

numbers a and b such that ab = 0, a = 0 and b = 0; the

axiom 2 is applied to the equality ‘‘ab = 0’’, formulated

with the two non-existing numbers.

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 cogni-

tive 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 & An-

tonini 2006).

The subject of the following protocol, Maria, is a uni-

versity student (last year of the degree in Pharmaceutical

Sciences) who is familiar with proof. In the following

excerpt of the interview, ‘I’ indicates the interviewer and

‘M’ indicates the student.

(1) I: Could you try to prove by contradiction the

following: ‘‘if ab = 0 then a = 0 or b = 0’’?

(2) 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.

(3) 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.

(4) I: Let us say that one can use that 0/b = 0.

(5) 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.

(6) I: Who does come back?

(7) 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 …(8) 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. […]

(9) I: [The interviewer shows the proof by contradiction

of the statement ‘‘ffiffiffi

2p

is irrational’’, then asks:] what

do you think about it?

(10) 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!

Maria is able to produce a proof, but she is doubtful about

its validity (2). The main difficulties emerge from stating

the validity of the sub-theorem. The cause seems to be the

upsetting of fundamental beliefs: Maria declares that she

lost the control on what is true and what is false (3). To

make herself clear, she distinguishes two ‘‘worlds’’: the

‘‘absurd’’ world and the ‘‘real’’ world (5). The ‘‘absurd

world’’ is the world where the false hypothesis of the

secondary statement is assumed. Similarly, the ‘‘rules’’

used in the proof of the sub-theorem belong to this ‘‘absurd

world’’; and since these rules are absurd too (5), they may

not coincide with the rules commonly applied to the ‘‘real

world’’.

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

406 S. Antonini, M. A. Mariotti

123

happen, including 0/b = 0. The absurdity of the hypoth-

esis 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’’ gener-

ated by the false assumption and in which the proof makes

sense.

The case of irrationality offfiffiffi

2p

is different. In this proof,

Maria does not feel that an ‘absurd world’ is involved,

because the fact thatffiffiffi

2p

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

The model highlights how a theorem with an indirect proof

involves two different theoretical levels: a crucial point

consists in the articulation of these levels. Moreover, the

management of the shift from the proof of the principal

statement to the proof of the secondary statement asks one

to move from the theory to the meta-theory where such a

shift can find validation (Table 2).

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-the-

ories, 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).

In other words, we formulate the hypothesis that the

proof of the secondary statement may not be intuitively

acceptable (in the sense of Fischbein 1987) as a proof of

the principal statement, as is commonly assumed. In the

following we analyze a protocol showing that the accept-

ability of the proof of the statement S* does not

immediately entail that the principal statement was proved,

even when the subject is able to describe in detail the

method of indirect proof. In the protocol, the interviewed

subject, Fabio, is a university student (last year of the

degree in Physics). He was asked to express his opinion on

the indirect proof (Fabio and the interviewer use the

expression ‘proof by contradiction’ to denote ‘proof by

contraposition’, as frequently happens in Italy).

(1) F: Proof by contradiction is artificial: how does one

get out of it? Ok, you have arrived to the contra-

diction… and then? […] I don’t see that conclusion

be linked to the other one, I miss the spark […]

(2) 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]

n2 = 2(2k2 + 2k) + 1 is odd.

(3) F: Yes, I understand, it is better to prove that if n is

odd then n2 is odd.

(4) I: And then, what is the problem?

(5) F: The problem is that in this way we proved that n

is odd implies n2 is odd, and I accept this; but I do

not feel satisfied with the other one.

(6) I: Do you agree that natural numbers are odd or even

and there are not other possibilities?

(7) F: Yes, of course... and now you will say: n2 is even,

n is even or odd, but if it were odd, n2 would be odd,

but it was even... yes, ok, I know, but… I’m not

getting something.

(8) 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.

(9) 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.

(10) 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

Table 2 Statements, proofs and theoretical levels involved in a the-

orem with indirect proof

Statements Proofs Theoretical levels

S* C

direct

T

theory

S*?S MP MT

Meta-theory

S (S*, C, T) + (MS, MP, MT)

indirect

T + MT

Theory and meta-

theory

Indirect proof: what is specific to this way of proving? 407

123

esclusione… cio che non e, non la cosa diretta. Va

tutto bene, ma quando mi devo ricollegare...’’]

Fabio clearly expresses his difficulty to grasp the link

between the contradiction and the validity of the principal

statement S (1): the source of difficulty seems to be the

meta-theorem.

Subsequently (2), the interviewer proposes a theorem

with a proof by contraposition. The principal statement is

S : if n2 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 n2 is odd.

Fabio makes explicit that what it is proved is the sec-

ondary 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). Never-

theless, 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 Fa-

bio’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 some-

thing‘‘, ‘‘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 argumenta-

tions 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 geom-

etry (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 conjec-

tures supported by an argumentation.

21 V: They [r and s] cannot be perpendicular because

otherwise it [line t] could be parallel to one of the two and not

intersecting the other one [he makes a drawing, see Fig. 1]

[…]

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 perpendic-

ular lines.

33 I: Then?

1 Valerio and Cristina do not belong to the same class, although they

belong to the same school.

408 S. Antonini, M. A. Mariotti

123

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.

Let us use our model to describe the whole process of

conjecturing and argumenting and to identify some of its

key elements. First of all, we observe that Valerio proposes

three conjectures:

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)

The second conjecture (S2) is a generalization of the first

one (S1), and the argumentations supporting S1 and S2 do

not have any significant differences, as the students say

(31–32). Both argumentations are indirect and it does not

seem that Valerio has any difficulties related to their

acceptability.

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 prin-

cipal statement (if A is true, r and s are not perpendicular/

crossing).

The case of the argumentation supporting S3 is dif-

ferent. 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 succes-

sive reformulations (33–40), and this process seems

fundamental. Different cases are considered (perpendicu-

lar, 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 argu-

mentation 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 argu-

ments 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, corre-

sponding 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 sponta-

neous 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, high-

lights 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?

Fig. 1 Valerio’s drawing

Indirect proof: what is specific to this way of proving? 409

123

After a phase of exploration, the students generated the

conjecture that the angle S (see the Fig. 2) is obtuse. Then

the interviewer asked them whether this angle might be a

right angle.

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 p/2 [right]

because it would become a quadrilateral.

The students formulate the conjecture that the angle S

cannot be a right angle, and they articulate the argumen-

tation in an indirect way. The argumentation produced can

be summarized as follows: if the angle is right then the sum

of two angles of the triangle is 180�, then the triangle

becomes a quadrilateral. After this argumentation, no

proofs are generated by the students.

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 Geo-

metrical 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 geo-

metrical 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 con-

tradiction. 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.

7 Conclusions

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 stu-

dents’ 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 diffi-

culties can be overcome. For example, the protocol of

Riccardo and Paolo reveals a meaningful different treat-

ment 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.Fig. 2 The angle between the two angle bisectors

410 S. Antonini, M. A. Mariotti

123

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.

First of all, by the tasks of producing and supporting a

conjecture, students can become aware of the different

activities involved in a theorem. This awareness is very

important in the specific case of indirect proof that is

sometimes refused because it is neither an efficient method

of discovery nor an explanatory argument. Let us consider,

for example, what Giacomo (last year of the degree in

Engineering) says:

‘‘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.’’

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’’).

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 acti-

vating these processes and then of constructing a bridge

to overcome the gaps that indirect proof seems to pro-

voke. On the contrary, without any conjecturing phase,

some gaps could not be bridged or could require sacri-

fices and mental efforts that not all the students seem to

be able to make.

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