the renaissance of abduction

Ferrando E.University of Genoa Department of Industrial Engineering and [email protected]

The Renaissance of Abduction

Introduction

In the article the reader will briefly travel from the birth of abduction as defined by Charles Sanders Peirce (1839-1914), father of the modern Semiotics, to the application of this concept in several fields like the Artificial Intelligence, Mathematics Education and many others.

This work concerns the problem of abductive processes in mathematics; more pre-cisely, in case of 18 years old freshmen who attend a Mathematical Analysis course for future engineers, and where they are asked to solve some open question problems. The work continues with the presentation of a cognitive model able to detect and describe creative processes of abductive nature, even in cases where the solver has to build the answer of the problem and not only has to find the hypotheses to justify it. The elements of such paradigm will be also employed to analyse the professor’s lecture in order to recognise the presence or absence of abductive attitudes.

The article ends with some considerations about the possible applications of the ele-ments of the cognitive model also from a didactical point of view.

Peirce’s Theory of Abduction

The philosophical foundation of Peirce’s work is to show that there is ‘a logic of discovery’, which is not to be confounded with the psychology, sociology and history of discovery.

Peirce writes: “each chief step in science has been a lesson in logic” (C.P. 5.363). His intention is that the birth of new ideas can never satisfactorily be cleared up by psychological, sociological and historical investigation alone. One important task of a philosopher is to conduct a logical (conceptual) investigation of discovery. There can be good reasons, or bad, for suggesting one kind of hypothesis over another. The reasons may differ entirely from those that lead one to accept a hypothesis. Peirces wishes to

Key words: Abduction, abductive system, creative processes, math-ematics, Peirce, problem-solving

Vol. 24 - n.3 (197-212) - 2009HUMAN EVOLUTION

What about abduction in mathematics tasks with no observ-able facts? Is it possible to talk of abductive processes in these cases? The Renaissance of Abduction during the last century and this last decade.

198 FERRANDO

show that reasoning towards a hypothesis is of a different kind than reasoning from a hy-pothesis. He realizes that the former “has usually been considered either as not reasoning at all, or as a species of Induction” (C.S. Peirce, Letters to Lady Welby); but he states: “I do not think the adoption of a hypothesis on probation can properly be called induction; and yet it is reasoning” (C.P. 8.388).

Many philosophers only concern themselves with analysing the reasons for accept-ing a hypothesis. Hanson (1959) notes: “They begin with hypothesis as given, as cooking recipes begin with the trout”. To study only the verification of a hypothesis leaves a vital question unanswered, namely, how hypotheses are ‘caught’. Natural scientists do not ‘start from’ hypotheses. They start from data. Peirce’s theory of abduction is concerned with the reasoning, which starts from data and moves towards hypothesis.

Peirce defines abduction as follows:

(…) Abduction is where we find some curious circumstances, which would be ex-plained by the supposition that it was a case of a certain rule, and thereupon adopt the supposition (…) (Peirce 2.624).

The surprising fact C is observed.However if A were true, C would be a matter of course.Hence, there is reason to suspect that A is true (CP. 5.188-189, 7.202)

C is true of the actual world and it is surprising, a kind of state of doubt we are unable to account for by using our available knowledge. C can be simply a novel phe-nomenon, or may conflict with background knowledge that is anomalous; A is a possible “explanation” of C. Therefore, abduction is any process employed to create or to call for a hypothesis aimed at explaining a fact.

Such definition can be schematised as follows:If H were true H → F therefore H is likely.

Furthermore,

ABDUCTION: Rule- All the beans from this bag are white Result- These beans are white Case- These beans are from this bag. (C.P. 2.623)

Which is different from

DEDUCTION: Rule- All the beans from this bag are white Case- These beans are from this bag Result- These beans are white

198

199AbDuCTION

INDUCTION: Case- These beans are from this bag Result- These beans are white Rule- All the beans from this bag are white.

According to Peirce there is a clear difference between abduction and induction: “The induction adds nothing. At the very most it corrects the value of a ratio or slightly modifies a hypothesis in a way, which has already been contemplated as possible. Ab-duction, on the other hand, is merely preparatory. It is the first step of scientific reason-ing, as induction is the concluding step (…). They are the opposite poles of reason, the one the most ineffective, the other the most effective of arguments. The method of either is the very reverse of the other’s (…). Abduction seeks a theory. Induction seeks for facts” (C.P. 7.217-218).

The renaissance of abduction

In the previous paragraph we briefly illustrated the definition of abduction accord-ing to Peirce; its birth has more than a hundred of years, and its study had a slow devel-opment, since logicians concentrated on deductive and inductive logic.

In the last decades, though, the importance of abduction in the discovery and evalu-ation theories has been recognized by philosophers of science. In the field of Artificial Intelligence abduction has become a key part of medical diagnosis and other tasks that require finding explanations. Lorenzo Magnani (2001) dedicated an entire book to this tenet. As Paul Thagard says:

Lorenzo Magnani’s new book contributes to this research in several valuable ways. First, it nicely ties together the concerns of philosophers of science and Artificial Intel-ligence researchers, showing, for example, the connections between scientific thinking and medical expert systems. Second, it lays out a useful general framework for discus-sion of various kinds of abduction. Third, it develops important ideas about aspects of abductive reasoning that have been relatively neglected in cognitive science, including the visual and temporal representations and the role of abduction in the withdrawal of hypotheses. The author has provided a fine contribution to the renaissance of research in explanatory reasoning.

In Mathematics and Mathematics Education, accounts of mathematics learning have long acknowledged the importance of autonomous cognitive activity, with par-ticular emphasis on the learners’ abilities to initiate and sustain productive patterns of reasoning in problem solving situations.

Lakatos (1976) acknowledged the nonlinearity of inferential reasoning, stating that “discovery does not go up or down, but it follows a zigzag path; prodded by counterex-

199

200 FERRANDO

amples, it moves from the naïve conjecture to the premise and then turns back again to delete the naïve conjecture and replace it with a theorem”.

Nevertheless, most accounts of problem solving performance have been explained in terms of inductive and deductive reasoning, containing little explanation of the novel actions solvers often perform prior to introducing formal algorithmic procedures into their actions. For example, cognitive models of problem solving seldom address the solver’s idiosyncratic activity, such as the generation of novel hypotheses, intuitions, and conjectures, even though these processes are seen as crucial tools through which mathematicians ply their craft (Anderson, 1995; burton, 1984; Mason, 1995).

Mason (1995) points out that in trying to avoid difficulties, “the curriculum turns everything into behaviour, avoids awareness, assumes deduction, tolerates induction, and ignores abduction”.

This last decade, though, has represented a period of change even in the mathemat-ics education field. Abduction has become an important topic to be considered. Cifarelli (1998, 1999) approaches abduction from the point of view of problem-solving strate-gies. He aims at clarifying the processes by which learners construct new knowledge in mathematical problem solving situations, with particular focus on instances where the learner’s emerging abduction or hypotheses help to facilitate novel solution activity. The basic idea is that an abductive inference may serve to organize, re-organize, and trans-form a problem solver’s actions.

Pedemonte (2003) makes a comparison between an abductive argumentation sup-porting a conjecture and its related proof. She stresses the importance of a structural analysis of a process which goes from an abductive argumentation to a deductive proof, and also from an abductive argumentation to an abductive “proof”.

De Hoyos (2004) is concerned with a study aimed at conceptualising how a group of undergraduate students tackle non-routine mathematical problems during a problem-solving course. The aim of the course is to allow students to experience mathematics as a creative process and to reflect on their own experience.

Peirce’s abduction, abduction in Artificial Intelligence; in Mathematics Education abduction as a tool to organize and transform problems solvers’ actions; furthermore, abduction in the construction of mathematical proof. What about abduction in cases of mathematical open question problems?

My personal research interest is related to creative processes of abductive nature in case of open question problems where the solver not only has to find hypotheses justify-ing a fact, but also has to look for a fact to be justified. This is the case of problems like the following one:

Given f differentiable function in R, what can you say about the following limit?

h

hxfhxfh 2

)()(lim 00

0

−−+→

201AbDuCTION

An a-priori analysis of this problem unearthed the insufficiency of Peirce’s defini-tion of abduction to frame and analyse potential students’ creative processes. Such in-adequacy is related to the manner in which Peirce’s abduction refers to the creation of a hypothesis that could explain an observed fact. In an abductive process a ‘starting fact’ is always considered and it is always true. On the contrary the aforementioned problem is an open-response task, which means a performance task where students are required to generate an answer rather than select it from among several possibilities, but where there is a single correct response.

The core of my work has become the understanding whether in cases of lack of observable facts it is appropriate to say that abduction (or something more general than Peirce’s abduction) could be one of the several kinds of processes students might adopt; and if such a kind of creative processes can be described by a cognitive model.

The construction of a new cognitive model

This study has been developed with the aim to build a broader and finer model (in comparison with Peirce’s definition of abduction) able to catch and to describe, with precise elements, a wider class of creative processes of abductive nature related also to those cases where the facts are not observable.

This construct, which I named Abductive System, does not contradict Peirce’s para-digm; on the contrary, it contains this last one.

The Abductive System contains four elements: facts, conjectures, statements, and actions. Referring to something as a fact in the Abductive System means to think it is true or correct. Facts are pieces of information that can be considered. In the Abductive System we say that a subject talks about or thinks of a fact when he or she is convinced of its truthfulness.

The term conjecture adopted in the Abductive System is meant as an opinion or judgement, a probable inference; a guess; or a suspicion. The definition is quite broad since the idea of conjecture in this context is not only like an assertion which seems to be true (fits all known cases) but for which there is as yet no definitive justification. Therefore, a subject refers to a conjecture when he or she is not fully convinced of its truthfulness.

In the Abductive System a conjecture owns two different roles:

the role of HYPOTHESIS, meant as an idea suggested as a possible explanation 1. for a particular situation or condition.

It is important, at this point, to clarify the difference between the term hypothesis used hare and the term abduction defined by Peirce. In the Abductive System the term abduction is meant as that cognitive process employed to look for or build possible ex-planations, or regularities under which a particular situation may have sense even when

202 FERRANDO

the fact is not already existing and true. It is that cognitive attitude which makes the solver asks himself: Are there conditions under which this fact makes sense? Can I find a hypothesis which could justify or explain such a situation? The term hypothesis in the Abductive System is the product of an abductive process. In Peirce’s theory of abduction the term hypothesis and abduction were used interchangeably to indicate the inferential process to find an explanation of a surprising already existing phenomenon, while the product of the inference was usually called CASE (see p.2).

The other role of the conjecture is that of

C-FACT (conjectured-fact). It has the ‘truth value” of a conjecture and it repre-2. sents a possible answer to the given problem or a possible answer to a certain step of the solving-problem process.

A C-FACT may turn into a fact if it is proved to be true, and it may become the answer to the problem, or an answer to a certain step of the solving problem process. A C-FACT could also be abandoned if it turns out to be “false”: in the sense the solver does not find any justifying hypothesis, or he finds a hypothesis proving its falsehood. Finally, a C-FACT that has been proved true, could be used twice; first, as a fact since its truthfulness has been proved, and then the new fact becomes a hypothesis in the sense it could be used to justify or explain a new C-FACT.

Facts and Conjectures are expressed by statements. In the Abductive System the statements are divided into the following categories:

Stable statements•unstable statements•

A stable statement is a proposition whose truthfulness and reliability are guaranteed, according to the individual, by the tools used to build or consider the fact or conjecture described by the proposition itself. Namely, the truthfulness depends directly on the tools employed in the construction phase (e.g.; a “visually-based” fact: the validity of the proposition describing the phenomenon is justified by a visual perception).

An unstable statement is a proposition whose truthfulness and reliability are not guaranteed, according to the individual, by the tools used to build or consider the con-jecture described by the proposition itself. Namely, the tools used in the creation phase are not sufficient for the solver to consider the conjecture described by the proposition as being definitively true. The consequence of this is the search of a hypothesis or an argumentation that might validate the aforementioned statement.

Of particular interest is the abductive statement, which is a proposition describing a hypothesis built in order to corroborate or to explain a conjecture. The abductive state-ments too, may be divided into stable and unstable abductive statements. The former, according to the solver, state hypotheses that do not need further proof; the latter require a proof to be validated, that means a process that brings back and forward.

It is important to clarify that the definitions of stable1 and unstable statement are student-centered, namely, the condition of stable and unstable is related to the subject:

203AbDuCTION

what can be stable for one student may represent an unstable statement for another stu-dent and vice versa; not only that, but the same subject may believe stable a particular statement at a certain point of his or her scholastic career, and this may become unstable later on when their base cultural knowledge of structured mathematical knowledge in-creases (e.g.; she or he learns new mathematical systems; new axioms and theorems). Furthermore, a stable statement may become unstable, inside a problem solving process, not because the student is convinced of that, but for a “cultural contract”; namely, at a certain point of the solving process the subject may recall his or her scholastic experi-ence and remember that a statement is considered stable if it is justified inside a precise mathematical system supported by axioms and theorems; thus the student will analyse the tools employed for verification if they satisfy such conditions. Another situation leading the student to reconsider a statement from stable to unstable is the “didactical contract”; the subject might believe the visual evidence to be sufficient in order to justify a conjecture, but the intervention of the teacher could underline its insufficiency and therefore the student would find himself or herself looking for new tools. Furthermore, the same statement may transform from unstable to stable inside a process itself because the subject follows the mathematicians’ path: they start browsing just to look for any idea in order to become sufficiently convinced of the truth of their observation, then they turn to the formal-theoretical world in order to give to their idea a character of reliability for all the community (Thurston, 1994).

The following example, taken from Harel’s Proof Schemes (1998) work, seeks to clarify part of this tension:

(…) Further, a person can be certain about the truth of an observation in one situ-ation, but seek additional or different evidence for the same observation in another situ-ation. For example, long before students learn geometry in school, they are convinced, based on personal experience and intuition, that the shortest way to get from one point to another is through the line segment connecting two points. Later, as participants in an Euclidean geometry class, an instantiation of this observation - stated in the theorem “The sum of the lengths of two sides of a triangle is greater than the length of the third side” – may become a conjecture for the students until they find evidence that would be accepted by their class community or their teacher. The kinds of evidence the students may look for are based on whatever conventions are accepted in their class as evidence for a geometric argument. These conventions may differ from one class to another; for example, what might be accepted as evidence in a standard high school Euclidean ge-ometry class is likely to be insufficient evidence for a college class studying axiomatic geometry (p. 243).

A fact is expressed by a stable statement since for the subject it is certainly true. Therefore, according to the Abductive System, can a stable statement, describing a fact, lead to an abductive process (meant as the search of justifying/explaining hypotheses of a situation)?

204 FERRANDO

No, it cannot, if the fact is the result of a transition from a C-Fact to a fact. The abductive process, though, could have been employed before, finding stable explaining hypotheses that allowed the transformation of the C-Fact into a fact.

Yes, it could, if the Fact expressed by the stable statement is something observed and considered by the subject certainly true but curious (remember the example of the fossils; or explorations made by the students using Dynamic Geometry Software like Cabri Geometre, for example); in this case the student could feel the necessity to look for those conditions under which such a fact does make sense. Other condition for a pos-sible abductive process in case of a stable fact is the didactical contract: the teacher asks ‘ Prove that…’ or ‘Explain why…’ even though the subject is convinced of the truthfulness of the fact and he is not surprised by it.

The character of instability expressed by an unstable statement may drive the sub-ject to search the way to transform the unstable statement into stable statement and this tension can generate an abductive process.

behind any statement there is an action. Actions are divided into phenomenic ac-tions and abductive actions. A phenomenic action represents the creation, or the “taking into consideration” of a fact or a c-fact: such a process may use any kind of tools; for example, visual analogies evoking already observed facts, a simple guess, or a feeling, “that it could be in that way”; a phenomenic action may be guided, for example, by a didactical contract or by a transformational reasoning2 (Harel, 1998).

An abductive action represents the creation, or the “taking into account” a justifying hypothesis or a cause, therefore the abductive action represents the abductive process meant in the broad way, namely the presence of abduction even in absence of an already observable fact; like the phenomenic action, they may be conveyed by a process of inte-riorization (Harel, 1998), by transformational reasoning (ibid) and so on. The abductive actions may look for: 1) a hypothesis, to legitimate the previous met or built conjecture; 2) a procedure, to legitimate or justify the previous built conjecture; 3) tools to legitimate the adaptation of an already known strategy to a novel situation.

After a broad description, the Abductive System could be schematised in the follow-ing way: conjectures and facts are ‘act of reasoning’ (boero, Ferrero, 1994) generated by phenomenic or abductive actions, and expressed by ‘act of speech’ (ibid) which are the statements. The adjectives stable, unstable, and abductive are not related to the words of the statements but to the acts of reasoning of which they are the expression. Hence, the only tangible thing is the act of speech, but from there, we may go back to a judgment concerning the act of reasoning thanks to the adjectives given to the statement.

Finally, for two different subjects the same statement may be stable or unstable. Therefore, two persons may achieve the same act of reasoning and judge it by a different method.

205AbDuCTION

Some applications of the abductive system

In this paragraph the author presents two different examples where the elements of the Abductive System are employed.

The first one is part of a broader experimentation conducted with a group of fresh-men enrolled in a required Mathematical Analysis course for engineers at the Italian Fac-ulty of Industrial Engineering and Management of Genoa (for more details see Ferrando 2005) during the academic year 2001-2002.

One of the problem posed to the students is Given f differentiable function in R, what can you say about the following limit?

previously mentioned in this article. At the time this task was proposed, the students have been exposed to the definition of differentiable function given through the limit of the difference quotient. Throughout the problem-solving process the students have been videotaped.

The second example concerns the analysis, through the tools of the Abductive System, of a part of a professor’s lecture in the same aforementioned Mathematical Analysis course.

Analysis of the protocolThe analysis of the protocol in this case is strictly related to the creative processes

of abductive nature, focusing only on the application of the elements of the Abductive System to catch and to describe abductive processes in cases of not observable facts.

The students were asked to work in pairs (they liberally choose whom to work with) ; such a choice was motivated by the conviction that the necessity of “thinking aloud” to communicate their own ideas gives the opportunity to bring to light guessing processes, creations of conjectures and their confutations; therefore, those creative processes which in great part remain “inside the mind” of the individual when one works alone, and very often only the final product is communicated to the others (Thurston, 1994; Lakatos, 1976; Harel, 1998).

It is important to note that the participants were not asked to produce any particular “structured” solution; the aim being to leave the students completely free to decide their solution process and to autonomously evaluate the acceptability of their solution by the teacher.

The following excerpt is taken from a pair of students (average level achieving): Daniele and betta.

h

hxfhxfh 2

)()(lim 00

0

−−+→

206 FERRANDO

D: in my opinion it is the same...when you do the limit of the difference quotient you do this over this (ndr: he signs the vertical segment and the horizontal segment. See the red segments in the figure below)

and ours would be this over this (ndr: he signs the vertical segment and the horizontal segment. See the red segments in the figure below)

(ndr: Daniele seems to be satisfied and ready to give me the answer).

Daniele looks at the graph and states that it is the same, referring to the standard difference quotient.We have a C-FACT (“it is the same thing”) created by a PHENOMENIC ACTION guided by a feeling, by a visual impact with the graphic representation of the standard difference quotient. The C-FACT is expressed by an UNSTABLE STATEMENT, since he does not believe the visual impact sufficient to validate his act of reasoning.

Creation of a HYPOTHESIS (“the same graphical frame”) through an ABDUCTIVE ACTION guided by the reinterpretation of the frame used for the standard difference quotient. Daniele translates the difference quotient as the ratio between the vertical and horizontal segments (see the two figures) and he shifts such interpretation to the present situation. The HYPOTHESIS is expressed by a STABLE ABDUCTIVE STATEMENT since the graphical justification results sufficient for him and satisfying for betta.He will tell me that since the two things are the same, then the limit will be the same, namely, the first derivate of f in x0.

EXCERPT ANALYSIS

Daniele sketches the graph of a function and marks the point x

0, x

0+h, x

0-h, f(x

0), f(x

0+h),

f(x0-h). (ndr: all the other marks made on the

graph are subsequent).

207AbDuCTION

Analysis of a part of a lectureLike in the case of the protocol (for reasons of space) the reader will just find the

analysis of the part of the lecture strictly related to the creative processes of abductive nature. The topic of the lecture is the proposal of some tasks regarding the continuity and differentiability of one-variable functions. The following table is divided into 3 columns. The first column contains the professor’s excerpt which is significant from an abductive point of view; the second column presents the interpretation through the tools of the Abductive System, if we look at it from the teacher point of view; in the last column the reader will find the interpretation through the tools of the Abductive System from the student point of view.

At the beginning the professor tackles the topic of the differentiability writing on the blackboard the definition through the limit of the difference quotient, and asking in a direct form which meaning such a kind of inscription may have. He continues the lesson with a graphic exploration of the meaning of the difference quotient and considering the change of such a graph with the introduction of the limit for h that goes to 0. In this phase the gestures, the use of graphical representations and the reference to them become fundamental tools. The basic idea is to link the expression

with its geometrical meaning, using the graphical visualization; and the further link between the first derivative and the formal expression of the tangent line. At this point the professor feels the necessity to reinforce graphically the idea of continuity, with the intention to give for each step, a sense that goes beyond the formalism.

ExcerptInterpretation through the tools of the Abductive System

For the Teacher For the Student

The first represents a discontinuous function, the second represents a continuous function.

FACT created by a PHENOMENIC ACTION guided by the need to make students understand the concept of continuity beyond the formal definition. It is expressed by a STABLE STATEMENT, because the teacher owns the cultural background that justifies such a fact.

C-FACT expressed by an UNSTABLE STATEMENT, because the visual impact should not be enough.

h

xfhxfh

)()(lim 00

0

−+→

208 FERRANDO

For small variations of x we have small variations of y.

Creation of a HYPOTHESIS through an ABDUCTIVE ACTION guided by the definition of continuous function. The hypothesis is stated by a STABLE ABDUCTIVE STATEMENT, because the definition seems to be enough to legitimate the hypothesis

Creation of a HYPOTHESIS through an ABDUCTIVE ACTION The hypothesis is stated by a STABLE ABDUCTIVE STATEMENT, because the definition seems to be enough to legitimate the hypothesis

The first function is differentiable everywhere, the second one is not differentiable everywhere

FACT created by a PHENOMENIC ACTION guided by the need to make students understand the concept of differentiability through its graphic meaning. It is expressed by a STABLE STATEMENT, because the teacher owns the cultural background that justifies such a fact.

C-FACT expressed by an UNSTABLE STATEMENT, because the visual impact should not be enough.

The second is not differentiable in the minimum value because in this it doesn’t have a tangent line, or better, in this point it has a tangent line that immediately changes theslope from this way \ to this / way

Creation of a HYPOTHESIS through an ABDUCTIVE ACTION guided by the relationship between differentiable functions and the geometrical meaning of the first derivative. It is expressed by a STABLE ABDUCTIVE STATEMENT, because the definition seems to be enough to legitimate the hypothesis

Creation of a HYPOTHESIS It is expressed by a STABLE ABDUCTIVE STATEMENT, because the definition seems to be enough to legitimate the hypothesis

209AbDuCTION

Well…a possible consequence of the derivatives, for example, is: if the first derivative is greater or equal to zero then the function is increasing (note: he writes the following formalization)f ’(x) ≥ 0 ⇒ f is increasing more precisely f ’(x) ≥ 0 ⇔ f is increasingif the function is differentiable (and he arranges in the following way)if f is differentiablef ’(x) ≥ 0⇔ f is increasing

is it clear? The theorem is obvious if you have understood what the derivative is: it is the slope of the tangent line

FACT created by a PHENOMENIC ACTION guided by the need to show the sense and the need of the first derivative. It is expressed by a STABLE STATEMENT, because the teacher owns the cultural background that justifies such a fact.

C-FACT expressed by an UNSTABLE STATEMENT, since for the student, so far, it is just the statement of a rule (theorem)

The first derivative is the slope of the tangent line. If the slope is positive that means that the function is going up (note: he imitates with his hands) if the slope of the tangent line is negative then the function goes down (note: again he shows it with the hands).

Creation of a HYPOTHESIS through an ABDUCTIVE ACTION guided by the relationship between the geometrical meaning of the first derivative and the graph of a function. It is expressed by a STABLE ABDUCTIVE STATEMENT, because the visualization of the dynamic behaviour of the function seems to be enough to legitimate the hypothesis

HYPOTHESIS. It is expressed by a STABLE ABDUCTIVE STATEMENT, because the visualization of the dynamic behaviour of the function seems to be enough to legitimate the hypothesis

210 FERRANDO

Conclusions

For a long time, cognitive models of problem solving seldom addressed the solv-ers’ idiosyncratic activities such as: the generation of novel hypotheses, intuitions, and conjectures, even though these processes are seen as crucial steps of the mathematician himself (Anderson, 1995; burton, 1984; Mason, 1995). Most of the problem solving performances in advanced mathematics were explained in terms of inductive and deduc-tive reasoning, and very little was the attention paid to those novel actions solvers often perform prior to their engagement in the actual justification process; although autono-mous cognitive activity in mathematics learning, and learner’s ability to initiate and sus-tain productive patterns of reasoning in problem solving situation, are issues considered important in the fields of mathematics research. In this last decade, though, the tenet of abduction rose to a new life and it has been tackled from different points of view: Arti-ficial Intelligence, Abduction in mathematics problem-solving strategies; Abduction and the structure of a proof. but what about abduction in cases of lack of observable facts? Is it possible to talk about abductive processes when the observed fact is not present? If so, is it possible to define a model describing creative processes of abductive nature, which involve not only the creation of the hypotheses, but also the birth of the facts?

The analysis of the protocols through the tools of the Abductive System (like the one proposed in this article) show the presence of this kind of process; also underlying that the Abductive System does not contradict Peirce’s paradigm but it contains this last one.

The analysis of the professor’s lecture, along with the analysis of the field notes taken by observing his lectures (see Ferrando, 2005) and the several hours spent talking with him, brought to light one of his most important aims, that is, making his students to understand how things work, especially from a geometrical point of view. The analysis through the tools of the Abductive System allowed to underline a particular teacher’s attitude adopted in some steps of his didactical transposition, when he wants to convey a creative process, which is already known by him, though. I called such approach an Abductive Scheme, to distinguish it from the definition of abductive process, as con-structed in this research. The process used in the didactical transposition can be defined as a “simulation of a creative process”, since the teacher already knows what to build and which hypotheses to use in order to validate or refute the constructed fact.

To clarify this point we could schematise the Abductive Scheme in the following way:

1ST STEP: proposal of an act of reasoning (boero, Ferrero, 1994)2ND STEP: for the teacher the act of reasoning (ibid) has the value of fact, since he

knows a-priori its truthfulness or falseness; the statement expressing the fact is therefore a stable statement. For the student the same act of reasoning becomes a c-fact, expressed therefore by an unstable statement and consequently needing a hypothesis validating or refuting it.

211AbDuCTION

In this way the professor also tries to avoid an Authoritarian Scheme (Harel, 1998) where the student uses, as validating justification, the assertion “it is true because the teacher said so.”

Finally the Abductive System seems to answer positively to the question if it is possible to detect creative processes of abductive nature, even in cases of lack of observ-able facts, since its elements consent to give a description of them. It also allowed to evidence those teaching styles which can enhance an “abductive atmosphere”, when the teacher does not just deliver the knowledge but he or she creates those conditions where the immediate creation of a fact entails “the necessity” to build or to look for a justifying hypothesis, generating in this way creative mechanisms.

This study leads to new further issues; for example, which cognitive elements may help the growth of an abductive aptitude, which kinds of students’ reasoning experiences may improve an abductive approach. From a didactical point of view, it would be inter-esting to study more in depth which kinds of teaching styles may enhance the abductive processes modelled by the Abductive System.

Notes

1 The concepts of stable and unstable are related, moreover, to the mathematical context. In Euclidean Geometry if a statement is stable, the problem will be only to find the tools to prove it. Namely, in Eu-clidean Geometry it is enough to find few variations of “targeted” drawings to guarantee the stability of a statement. In Arithmetic the problem is more complex; it is sufficient to think of Goldbach’s con-jecture. Goldbach’s original conjecture (sometimes called the “ternary” Goldbach conjecture), written in 1742 in a letter to Euler, states “at least it seems that every number that is greater than 2 is the sum of three primes”. Note that here Goldbach considered the number 1 to be prime, a convention that is no longer followed. As re-expressed by Euler, an equivalent form of this conjecture (called the “strong” or “binary” Goldbach conjecture) asserts that all positive even integers ≥ 4 can be expressed as the sum of two primes. Not only a proof has not been found yet, but also, even though many millions of even numbers have satisfied such property, we are still not sure of its validity.2 “Transformational observations involve operations on objects and anticipations on the operations’ results. They are called transformational because they involve transformations of images-perhaps ex-pressed in verbal or written statements- by means of deduction” (Harel, 1998. p.258)

References

Anderson M., 1995, Abduction, Paper presented at the Mathematics Education Colloquium Series at the university Of North Carolina at Charlotte, Charlotte.

boero P., Ferrero E., 1994, Interplay between classroom practice and theoretical investigation: a case study concerning hypotheses in elementary mathematical problem solving, Proceedings of the 5th International Conference on Systematic Cooperation between Theory and Practice in Mathematics Education, Grado, Italy: 35-45.

212 FERRANDO

burton L., 1984, Mathematical thinking: the struggle for meaning, Journal for Research in Mathemat-ics Education, 15 (1): 35-49.

Cifarelli V., 1998, Abduction, Generalization, and Abstraction in Mathematical Problem Solving, Paper prepared for the annual meeting of the Semiotic Society of America, Toronto, Canada.

Cifarelli V., 1999, Abductive Inference: Connection Between Problem Posing and Solving, Proceedings of PME – XXIII, Haifa, 2: 217-224.

Fann K.T., 1970, Peirce’s Theory of Abduction, Martinus Nijhoff-The HagueFerrando E., 2005, Abductive Processes in Conjecturing and Proving, Ph.D. Thesis, Purdue university,

West Lafayette, Indiana. uSA.Harel G., Sowder L., 1998, Students’ Proof Schemes, Research on Collegiate Mathematics Education,

Vol. III, E. Dubinsky, A. Schoenfeld, & J. Kaput (Eds.), American Mathematical Societyde-Hoyos M. G., 2004, Solutioning : A Substantive Theory of Students’ Problem Solving Processes, The

Grounded Theory Review, 4 (1): 59-85.Lakatos I., 1976, Proofs and Refutations: The Logic of Mathematical Discovery, Cambridge, Cam-

bridge university Press.Magnani L., 2001, Abduction, Reason, and Science. Processes of Discovery and Explanation, Kluwer

Academic/Plenum Publishers.Mason J., 1995, Abduction at the heart of mathematical being, Paper presented in honour of David Tall

at the Centre for Mathematics Education of the Open university, Milton Keynes, uK.Pedemonte b., 2003, What kind of proof can be constructed following an abductive argumentation?,

Proceedings of the Third Conference of the European Society for Research in Mathematics Educa-tion, bellaria, Italy, 28 February – 3 March 2003.

Peirce C. S., Collected Papers. Volumes 1-6 edited by Charles Hartshorne and Paul Weiss. Cambridge, Massachusetts, 1931-1935; and volumes 7-8 edited by Arthur burks, Cambridge, Massachusetts, 1958.

Peirce C.S, 1953, Letters to Lady Welby, Irwin Lieb, ed. New York, Whitlock’s: 42Thurston W.P., 1994, On Proof and Progress in Mathematics, bulletin of the American Mathematical

Society, 30 (2): 161-177.