the relevance of Peircean theory

18

T H E R E L E V A N C E OF P E I R C E A N T H E O R Y OF

ABDUCTION TO T H E D E V E L O P M E N T OF STUDENTS'

CONCEPTIONS OF P R O O F ( W I T H P A R T I C U L A R

A T T E N T I O N TO P R O O F IN C A L C U L U S )

Elisabetta Ferrando Purdue University

The purpose of this paper is to illustrate the concept of abduction as a very important tool to explain and interpret mathematical inferences, with particular attention to the process of making proofs in calculus. The intro­duction is followed by a brief overview of Peirce's theory of abduction. The relationships between abduction and mathematics have been analyzed in or­der to explain why abduction can be considered relevant to the development of students' conceptions of proof. Finally, further questions for future re­search are raised.

Introduction

Deduction and induction have always been considered the two tradi­tional inferential types of reasoning. Yet, they seem to be not enough to ex­plain the inferential processes mathematicians generate in the act of making a proof.

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The philosopher and logician Charles S. Peirce asserted that another pattem of reasoning occurs in both science and everyday life called abduc­tion. According to Peirce, abduction is not only a creative act of making up explanatory hypotheses but also is a way of deciding for or against given hypotheses.

hi this paper the concept of abduction has been used to illustrate its deep connection with mathematical reasoning, to sustain the idea that it rep­resents a very important tool to study students' conceptions and misconcep­tions of mathematical proofs, to build an interpretation of the mechanisms students use to generate a proof, and in the future it could be used to build a whole model that can explain which kind of cognitive processes students use to make proofs in calculus.

A Brief Overview of Peirce's Theory of Abduction

Peirce asserted that there occurs in science and everyday life a pattem of reasoning wherein explanatory hypotheses are constmcted to account for unexplained data or facts. Peirce called this kind of reasoning abduction, distinguishing the process from the two traditionally recognized inferential types of reasoning, induction and deduction. Furthermore, Peirce saw the process of abduction not only as a creative act of making up explanatory hy­potheses but also as an evidencing process, namely, a way of deciding for or against given hypotheses.

Peirce argued that induction, deduction, and abduction represent distinct stages of inquiry. According to Peirce, a deductive inference is an analytical process by which particulars follow from general premises; thus, no new knowledge is produced in the process since there is nothing in the particular cases that is not implied by the premises, hiduction and deduction alone cannot account for the introduction of new knowledge without abduction:

B y induction we conclude that facts, s imilar to observed facts, are tme i n

cases not examined. B y abduction, we conclude the existence o f a fact

quite different f rom anything observed, f r o m which , according to k n o w n

laws, something observed would necessari ly result. T h e former, is reason­

ing from particulars to general laws; the latter, from effect or cause. T h e

former classif ies , the latter explains. (Peirce 2 .636)

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Using the same three propositions, the differences between deduction, induction, and abduction may be illustrated as follows:

Deduction

RULE: all the beans from this bag are white

CASE: these beans are from this bag

.-.RESULT: these beans are white

Induction

CASE: these beans are from this bag

RESULT: these beans are white

.-.RULE: all the beans fron this bag are white

Abduction

RULE: all the beans from this bag are white

RESULT: these beans are white

.-.CASE: these beans are from this bag

Thus, deduction is nothing but the application of a rule. Induction is the inference of the rule from the case and the result; name­

ly, it is where we generalize from a number of cases of which something is true, and infer that the same thing is true of a whole class. Or, where we find a certain thing to be true of a certain proportion of cases and infer that is true of the same proportion of the whole class.

Abduction is where we find some very curious circumstances which would be explained by the supposition that it was a case of a certain general rule, and thereupon adopt the supposition. For example, "fossils are found; say, remains like those of fishes, but far in the interior of the country. To explain the phenomenon, we suppose the sea once washed over this land". This is an abduction (Peirce 2.624).

The main difference between induction and abduction is that the former infers the existence of phenomena such as we have observed in cases which are similar, while abduction supposes something of a different kind from what we have directly observed, and frequently something which would be impossible for us to observe directly. According to that, Peirce produced the

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following example: "numberless documents and monuments refer to a con­queror called Napoleon Bonaparte. Though we have not seen the man, yet we cannot explain what we have seen, namely, all these documents and monuments, without supposing that he really existed. Hypothesis again" (Peirce 2.625).

The following chart may synthesize the possible relationship between induction and one of the aspects process of abduction:

F fact

H hypothesis

if H true H => F so H is likely

F H=>F

H

as i n d u c t i o n it is the inference of the RULE from the CASE and the RESULT. In the i n d u c t i o n F is a special case of H. ex. CASE (minor premiss) ~ The beans are

from this bag. RESULT - These beans are white. .*. R U L E (major premiss) - All the beans

from this bag are white.

The "kind" of the statement of the Minor Premiss and of the Major Premiss is the same: the "whiteness"

as a b d u c t i o n it is the inference of a CASE from a R UL E and a RESULT. Better, it is the acceptance of a minor premiss as a hypothesis on the strength of its "fittingness" to a known major premiss and a factual conclusion. In the a b d u c t i o n F is oQt a special

case of H. ex. R U L E (major premiss) - All the

beans from that bag are white.

RESULT - These beans are white. .*. CASE (minor premiss) - The

beans are from this bag.

The "kind" of the statement of the

Minor Premiss and of the Major

Premiss is different: "whiteness" vs.

"coming from the same bag".

Abduction as Idea Creation

SEMIOTICS OF EDUCATION/FERRANDO 22\

According to Peirce, abduction involves reasoning from the known (the rule) to the new or unknown (the case) by way of metaphoric projection, and the ideas represent the instruments of this connection.

He argues that "The elements of every concept enter into logical thought at the gate of perception" (5.131). Judgment enters in at the second and third stages of idea construction, as one moved beyond considering what is possible and attempts to actually test the feasibility of the new idea and ascertain how it fits with a series of related ideas.

As a communicative act, abduction involves a sign (the metaphoric in­stantiation of meaning), an interpretant (the set of experiences one antici­pates having in relation to an object), and the object one wishes to under­stand.

The metaphoric process involves a blending of all three of the sign functions — icon, index, symbol.

According to Peirce, signs are related to objects by resembling them, being causally connected to them, or being conventionally tied to them. Iconic signs are used for resemblance, indexical for causal connection, and symbolic for conventional association (Berger 1999).

Berger, in Signs in Contemporary Culture an Introduction to Semiotics, provides the following chart that can be useful for clarifying these defini­tions:

Sign Icon Index Symbol Signify Resemblance Causal Cormection Conven­by: tion Exam­ Pictures Smoke/fire Words ples: Statues of great Symptom/disease Gestures

figures Red spots/measles

Process Can see Can figure out Must leam

I

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Coming back to the whole process, it can be so described: it begins with the "drinking in" of an impression (Peirce 2.134-35). Attention soon shifts to how and in what ways the new idea sheds light on the referent, hi the following step, the focus shifts once again to the attempt to generalize and systematize the knowledge that has been gained.

Icons or images represent the form taken by the ideas when they first originate, as mere possibilities. Peirce called this first impressionistic pro­cess immediate interpretant.

At the second stage, defined by Peirce as the dynamic interpretant stage, the focus shifts to how well the sign delivers on its initial promise.

The final step is represented by the logical interpretant stage, when attention tums to a broader set of issues. The question becomes one of how well the new idea relates to other, similar ideas. The sign-object relationship at this stage is described by Peirce as symbolic in nature. At the symbolic stage, individuals attempt to locate or situate new ideas in a system of relat­ed ideas. The process of negotiation at this moment is particularly impor­tant.

It is possible to claim that Peirce falls under the social constmctivist m-bric, because he assigns a high priority to social interaction in helping indi­viduals formulate and test ideas.

Abduction and Mathematics

Mathematicians and mathematics educators have recognized the influence of abductive processes in mathematical thinking, although under different names. Lakatos (1976) acknowledged the nonlinearity of inferen­tial reasoning as he says; "discovery does not go up or down, but it follows a zig-zag path; prodded by counterexamples, it moves from the naive con­jecture to the premises and then tums back again to delete the naive conjec­ture and replace it by a theorem" (46).

Mason (1995) points out that in trying to avoid difficulties, "the cur­riculum tums everything into behavior, avoids awareness, assumes deduc­tion, tolerates induction, and ignores abduction" (4).

Accounts of mathematics leaming have long acknowledged the inpor-tance of autonomous cognitive activity, with particular emphasis on leam-

SEMIOTICS OF EDUCA TION / FERRANDO

ers' ability to initiate and sustain productive pattems of reasoning in prob­lem-solving situations. 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 cmcial tools through which mathematicians ply their craft (Anderson 1995; Burton 1984; Mason 1995).

Abduction and Creativity

The forming of an explanatory hypothesis requires creativity and sup­plies the leamer with ideas from which to make conjectures about potential courses of action to carry out. The importance of such creative activity has been emphasized by Burton (1984: 38), identifying the process of making conjectures as a component of mathematical thinking through which "a sense of any underlying pattem is explored".

To claim that abduction is a creative logical process is to say that the initial conceiving of a novel hypothesis is a causal effect of perceptions that occur in the course of the purpose-generated process of abductive reason­ing, not that this conceiving come into being as a result of any inference within the abductive process.

By a hypothesis, I mean not merely a supposition about an observed ob­ject, as when I suppose that a man is a Catholic priest because that would explain his dress, expression of countenance, and bearing, but also any oth­er supposed tmth from which would result such facts as have been observ­ed, as when van't Hoff, having remarked that the osmotic pressure of one percent solutions of a number of chemical substances was inversely pro­portional to their atomic weights, thought that perhaps the same relation would be found to exist between the same properties of any other chemical substance. The first step of a hypothesis and the consideration of it, whether as a simple interrogation or with any degree of confidence, is an inferential

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step which I propose to call abduction in keeping with Peirce's use of the term.

How can Abduction be Relevant to the Development of Students' Conceptions of ProoP

A mathematical proof, I believe, is a dynamic process which begins with a creative act, with the conception of several hypotheses which could explain the observed fact. This is exactly what abduction is about.

Very often, as a consequence of educators' erroneous approaches, stu­dents develop several misconceptions about mathematical proofs. The most common misconceptions are:

— a mathematical proof has a fixed schematic pattem that has to be fol­lowed, and where one's own creativity, and personal effort cannot fmd space;

— the stmcture of any theorem students experience is represented by hy­pothesis then thesis. The consequence of this fact could be the following misconception: the thesis cannot be conceived before the hypothesis. Again, an abductive approach in constmcting a proof could help students to understand that thesis and hypothesis are strictly interwoven, and very often the thesis is to be the first step, because it represents the "surprising observed facf that needs a hypothesis to be explained.

To investigate students' ideas about mathematical proof, the following questions could be asked: "How do you describe the process of making a proof? Give specific examples from calculus to illustrate your point". "Do you think there is a 'chronological' sequence in the construction of the hy­potheses and thesis of a mathematical theorem?"

I f we consider, for example, the geometrical interpretation of the first derivative, it is possible to see how iconic representations and indexical signs are involved in leading the idea of the relationship between the first derivative and the slope of a tangent line.

A second example might be represented by the following problem: let f and g be two functions which are defined on R. f is supposed strictly de-

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creasing in R, and g is supposed strictly increasing in R. What can we say about the function m(x)=f(g(x))? Is it increasing in R? Is it decreasing in R? Is it neither decreasing nor increasing? Prove it.

In this case the subject is asked to come up with a conjecture: the in­crease or decrease of the function and to prove it.

How wil l the student construct his conjectures? Which role wi l l the mental images and metaphorical reasoning have in such a process of prov­ing? Which kind of sign (intended as metaphoric instantiation of the mean­ing) wil l be used?

Which kind of experiences wil l the subject anticipate that wi l l help him in the process of proving? How wil l icons, indexes, and symbols be used?

The Application of Peircean Theory of Abduction in Reinterpreting a Proof Process

Hard and Sowder (1998), in "Students' Proof Schemes", define the act of proving as follows:

B y proving we m e a n the process employed by an individual to re­

m o v e or create doubts about the tmth o f an observation. T h e process o f

proving includes two sub-processes: ascertaining and persuading. Ascertaining is the process an individual employs to remove her or his

o w n doubts about the tmth o f an observation.

Persuading is the process an individual employs to remove others'

doubts about the tmth o f an observation. Therefore , a person's proof

scheme consists o f what constitutes ascertaining and persuading for that

person. (26)

According to Harel and Sowder, three different main categories exist: Extemal Conviction Proof Schemes, Empirical Proof Schemes, and Analy­tical Proof Schemes. The Transformational Proof Scheme belongs to the third category and it is so defined: "Transformational observations involve operations on objects and anticipations of the operations' results. They are called transformational because they involve transformations of images — perhaps expressed in verbal or written statements—by means of deduction" (30).

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What follows is an example of how the Peircean theory of abduction can be applied to reinterpretthe process of transformational reasoning. Both processes are shown below.

Amy demonstrated to the whole class how she imagines the theorem, "The sum of the measures of the interior angles in a triangle is 180^". Amy said that she imagines the two sides AB and AC of a triangle ABC being rotated in opposite directiais through the vertices B and C, respectively, until their angles with the segment BC are 90^ (figures la, b). This action transforms the triangle ABC into the figure A ' B C A " , where A ' B and A ' 'C are perpendicular to the segment BC. To recreate the original triangle, the segments A ' B and A"C are tilted toward each other until the points A ' and A " merge back into the point (figure 1 c). Amy indicated that in doing so she "lost two pieces" from the 90^ angles B and C (i.e., angles A ' B A and A ' 'CA) but at the same time "gained these pieces back" in creating the angle A. This can be better seen i f we draw AO perpendicularto BC: angles A ' B A and A"CA are congruentto angles BAO and AOC, respectively(figure Id).

B a C B b C B c C B d C

Figures la-d.

Harel's interpretation of this procedure is the following: Amy views a triangle as a dynamic entity; it is a product of her own imaginative construc­tion, not of a passive perception. Her operations were goal oriented and intended the generality aspect of the conjecture. She transformed the trian­gle and was fully able to anticipate the results of the transformations,name-ly, that the change in the 90^ angles B and C caused by the transformations is compensated for by the creation ofthe angle A. A l l this leads to her de­duction that the sum of the measures of the angles of the triangle is 180^.

SEMIOTICS OF EDUCATION/FERRANDO 111

At this point we might reconsider Amy's proof trying to interpret her process using abductive reasoning.

Amy knows that the sum ofthe angles ofthe figure lb is 180"" (it be­comes the rule)\ what she has to prove is that the sum of the angles of a tri­angle is 180"̂ (it becomes the result). At this point she makes the following hypothesis:

i f

J J

thena + 6 + A = 180°

With the abduction scheme it would be:

R U L E : = 180°

R E S U L T :

C A S E :

a + b + X = 180°

Therefore, her aim is to prove that the triangle is the same as the open rectangle, and doing so she proves that the sum of the angles of the triangle is 180^

In other words. Amy is in front of the "surprising fact" that the sum of the angles of a triangle is 180"̂ (result); at this point she tries to make up a hypothesis (case) that i f it is likely can explain the observed fact.

I

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Then, how can abduction better capture the process of transformational reasoning?

Transformational reasoning is a dynamic process that involves transfor­mations of concrete objects (for example, symbols) and mental images (for example, icons); it operatesby anticipatingthe results of the transformations (for example, the construction of hypotheses), and it uses several means of communication (drawings, mathematical symbols, verbal or written state­ments). Therefore, creativity, dynamicity, and a zig-zag path are all fea­tures that characterize an abductive approach.

What follows is a series of Calculus problems that could be used in a clinical interview with students for the purpose of investigating students'ab-ductive/transformational reasoning:

Problem #1: Letfbe a function which is continuous everywhere on a closed interval [a,bj and which has a derivative at each point of the open interval (a.b).

Also, assume that f(a)=f(b) then there is at least one interior point c of (a,b) such that f (c) = 0.

Problem #2: The following limits are of basic importance. Convince your­selfand then your peers that in none of these cases is it even easy to see that the limit does exist.

Problem #3: Would you expect that all linear functions are continuous? Justify your answer.

Problem #4: The functions f and g are related by g(x) -f(x)-^7

How are the graphs related? How is the slope of the graph o f f a t (a, f(a)) related to the graph of g at (a, g(a))? Why? Generalize your answers if you can.

SEMIOTICS OF EDUCATION/FERRANDO 229

The first problem represents the proof of Rolle's theorem. The follow­ing features could be encountered:

1. Connection with a geometric representation: the first derivative equal to zero implies a horizontal tangent (then, a link with a transformational proof scheme).

2. The student might start his proof using a constant function and show that it "works". Then, the student might create a hypothesis like "all con­stant functions satisfy the problem" and add further extensions to all con­tinuous functions with the aforementioned characteristics (therefore, cre­ating a hypothesis abductive process).

3. Transformation of algebraic symbols to geometrical representations and vice versa (transformational proof scheme).

4. Creation of other kiads of hypotheses (abductive process).

Problem 2 requires:

1. Creation of a hypothesis to explain why is not easy to see that such limits exist (abductive process).

2. Transformation of the algebraic symbols into other algebraic symbols or visual representations like the graphs of f(x) = sinx and f(x) = x (trans­formational reasoning).

Problem 3 leads students to a total creative process. They previously didn't know i f all linear functions are continuous or not. They have to con­struct a conjecture on their own and then substantiate it. The question seems to favor processes like: creation of hypotheses, evidencing arguments, trans­formational reasoning.

hi problem 41 assumed that students don't know any formula related to shifting a graph of a function. A possible strategy could begin by conjectur­ing what happens with a particular function, and then trying to generalize. Another possibility could be the use of a general function from the begin­ning.

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As in the other cases, transformational reasoning, construction of con¬jectures and evidencing processes seem to be necessary.

Further Considerations

The common feat-ures of these exercises are represented by the likely need of transformational reasoning and abductive approaches. Neverthe­less, each of them represents a different level of difficulty in the application of these "cognitive tools".

In problem 1, the students are already given the conj ecture that the have to prove: there is at least one interior point c of (a,b) such that f(c)=0.

In problem 2, the question is more difficult because the subjects are asked to find out the reason of the difficulty to see that the limits do exist.

In problems 3 and 4, students are not given any conjecture and conclu­sion. They must count on their own creativity to make hypotheses, draw con­clusions and eventually generalize the results.

Conclusions

Everything that has been discussed in this paper represents part of the theoretical framework for my doctoral research that wi l l start in the field at the beginning of the next year. My purpose is to build a possible model that explains which kind of cognitive processes students use to make proofs in calculus. The study wi l l be conducted according to my following personal beliefs:

The act of proving is a creative act.

The cause of most students' misconceptions about proofs is due to the way mathematical proofs are taught.

There is a big gap between the mathematician's activity of proving and the one that is a taught to our students.

SEMIOTICS OF EDUCATION/FERRANDO 231

Usually students' potential capacity of proving is better at the beginning of their school career. With "scholarization" the student loses part of the mathematician's quality of proving.

The questions so far which wil l lead my inquiry are:

Assuming that the process of making a proof starts with a conjecture, namely a constmction of a hypothesis, is it possible to delineate a model which explains how these conjectures take place, and how they are trans­formed during the proving process?

Is abduction a "natural" approach to a proving process?

Peirce talks about metaphors, icons, index, symbols in the process of ab­duction. Is the nature of the constmction of the ideas/conjectures meta­phorical? Are the first icons students come up with metaphorical or do they resemble more the subject they have to prove?

Which kind of icons, in calculus proofs, do students constmct that then be­come indexes which end up being symbols? Is it a linear development or is it more of a zig-zag process?

The entire research wil l involve a group of engineering students at the University of Genova(Italy). The inquiry wi l l last, probably, one year dur­ing which interviews with students, observations in class, field notes, vide­otaping, and analysis of their works on particular assigned problems wil l constitute the sources of the data.

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