f. arzarello - l. bazzini - g.p. chiappini intensional … · 2020. 9. 20. · 110 e arzarello - l....

Rend. Sem. Mat. Univ. Poi. Torino Voi. 52, 2 (1994)

WALT1

F. Arzarello - L. Bazzini - G.P. Chiappini

INTENSIONAL SEMANTICS AS A TOOL TO ANALYZE ALGEBRAIC THINKING

Abstract. The paper deals with some major difficulties usually met by students in learning elementary algebra. In order to understand them, the authors elaborate a theoretical frame, which uses key-concepts taken from logie and linguistics.

The frame is shown in action analyzing the processes of solution of an algebraic problem given to undergraduate students.

Introduction

It is very well known that in many approaches to algebra, formai aspeets of manipulations are at the core of didactical interventions. In many cases, as some researches point out (i.e. see Sfard [92], Kieran [91], Drouhard [92]), it is the very way in which students interpret formulas to cut off semantical aspeets and to emphasize only the syntactical ones: as a consequence, the potentiàl of a new body of knowledge remains detached by previous ones.

Existing literature shows that it is relatively easy to take snapshots of such problems in teaching and learning Algebra, but that it is very difficult to get tools of analysis able to suggest concrete didactical interventions (for example, see the volume Wagner & Kieran [89] and the discussion in Bell, Malone & Taylor [87]).

This paper attempts an analysis of algebraic thinking as observed in classroom experiments, from a (partially) new point of view, namely intensional semantics. Although no definitive conclusions can be drawn at the present state of research, we take into. account some basic questions concerning algebraic learning from a didactical and from an epistemological point of view.

The paper is divided into five parts, which consist of:

1. Landscape: ingredients and dimensions of algebraic reasoning are put forward.

2. Typical mistakes and misconceptions: the main problem of algebraic thinking is isolated as a problem of interpretatìon.

106 E Arzarello - L. Bazzini - G.P. Chiappini

3. Syntax and semantics of algebraic thinking: the main points of intensional semantics are sketched with respect tp algebraic language; the transformative aspects of algebraic language are scrutinized and interpreted using intensional machinery.

4. Some problems of algebraic learning and some typical misconcepts of students are described using the intensional paradigm: algebraic learning emerges as a game of interpretation.

5. Finally, we draw some remarks on pupils' errors according to the proposed framework.

1. The three and a half main dimensions of algebraic thinking

Algebraic thinking is here considered as developing along four main axes, three of which are opposed polarities, while the fourth is a scale of levels. More precisely, we put the polarities naturai language vs/ symbolic writing in the z-axis; sintax vs/ semantics in the ?/-axis; procedural vs/ relational in the z-axis. In the u>-axis we put the different levels of abstraction at which the symbolic expressions of algebra are constructed, interpreted and used (fig. 1).

Relational

A Naturai language

Symbolic writing

Fig. 1

Intensional semantics 107

Thinking algebraically means monitoring x,y, z polarities at a high w-level;

otherwise algebraic thinking is not completely mastered and sometimes is replaced by

surrogates (fig.2).

NATURAI. LANGUAGE

written arithmetic [procedurali

orai orai arithmetic syncopate algebra [procedural]

[procedural]

written syncopate algebra

[procedural]


In the end, there is a scale of levels of abstraction, where the different processes can be located (w-axis).

2. The interpretation of algebraic expressions

It is likely that diffìculties with algebra derive from the fact that students do not monitor and do not integrate ali the polarities listed in § 1. In addition they use formai objects at a very low level of abstraction. For ex., the pseudostructural student described in Sfard [91b] cuts off semantic dimension and reveals to be "unable to imagine the intangible entities (functions, truth sets) which he or she is expected to manipulate, [...] and would use pictures and symbols as a substitute: an algebraic formula, the variable, the letters A and R, each of these signs will turn into a thing in itself, not standing for anything else" (Sfard [92], p.2).

In our view, there are two reasons at least, which prevent students from using symbolic expressions as thinking objects:

a) The first is widely acknowledged: pupils who are not able to cope with the algebraic symbolism as a mean for reasoning prefer solution strategies rooted in semantics, mostly related to naturai language and procedural knowledge. For ex., novices faced with simple word problems fìrstly solve the problem by means of orai arithmetic and then attempt to translate their strategies and processes into equations (see the discussion in Kieran [89], p.35 ff). Additional examples are given by Laborde [82]): 12-14 y.o. students who are asked to answer questions like: "think a number, subtract 10 from it.Jhen add.Jhen multiply...Now think another number and do the sanie. Doyou get always the sanie result? Why?" do not use formulas but replace algebra with extralinguistic tools. This means that a student may be able to solve problems concretely in some cases, but her/his thinking tòols are not the symbolic expressions of algebra.

The existing literature has widely described the inhibition of algebraic machinery as a thinking tool: many research studies have pointed out the epistemic, didactic and cognitive obstacles preventing the learning of algebra. There is evidence that they can be generated by Arithmetic - as a body of organized operative knowledge integrated in the student's culture and language - or by a stack of problem fields and rules, which are taught in the classroom, for ex. according to old-aged tradition; età. In short, the first diffìculty for pupils who solve an algebraic word problem is the rising of a gap between arithmetic operative knowledge (where primary processes have been built, see Sfard [91a, 91b, 92]) and the higher level of algebraic knowledge (secondary processes, ibid.). It is hard for the student to interprete a problem into the new language and it is hard for the teacher to motivate this necessity (for the latter see Chevallard [89]).

b) The second reason is related to the very meaning of algebraic symbolism, namely


the interpretetion of letters as variables and parameters in the full sense of modem algebra after Viète. This has been emphasized by Lagrange as follows: "Ce qui la distingue [l'algebre] essentiellement de l'arithmétique et de la geometrie consiste en ce que son objet n'est pas de trouver les valeurs mèmes des quantitès cherchées, mais le systènte d'opérations àfaire sur les quantitès données pour en dèduire les quantitès qu'on cherche, d'après les conditions du problème. Le tableau de ces opérations est ce qu'on nomme en àlgebre une formule; et lorsqu'une quantité dèpend d'autres quantitès, de manière qu'elle peut ètre exprimée par une formule qui contieni ces quantitès, on dit alors qu'elle est une fonction de ces mèmes quantitès; de sort qu'on peut definir l'algebre, l'art de determiner les inconnues par des fonctions de quantitès connues, ou qu'on regarde cornine connues" (quotation from Chevallard [89], p. 158; underlines are of the authors).

The point here is that many times such a full sense may be narcotized by the type of problem and by the consequent request of translatìon into equations. For example, word problems whose corresponding equation is of the type ax 4- b = 0 (a, b specifìc numbers), admit a one-to-one translation of arithmetic computations, which is sufficient for getting the solution. At this regard, see Filloy & Rojano [89], who have compared this case with the case o>x 4- b — ex 4- d, where such one-to-one translation from arithmetical computations to algebraic formulas is no longer possible. On the other hand, as widely discussed in Chevallard [89], didactical situations requiring the use of letters as parameters are very rare in school. Ali this leads to strengthen the meaning of letters as unknowns and not as variables.

Cases a) and b) show that one of the main problems in facing algebraic learning is given by questions of interpretation. This is obvious in the first case, where algebra shows to be like a foreign language into which the problem situation must be translated. In the second case, it is interesting to observe that the meaning of Lagrange quotation consists in the fact that to make algebra means essentially to study V3-formulas, that is formulas like VaV6Vc3a:3y [ax 4- by = e] and not only to consider formulas like 3x[3x 4-5 = 0] (3-formulas, e.g. numerical equations, variable as unknown) or like Va[2a = a 4- a] (V-formulas, e.g. identities, variable as generalized number). In fact, while the formulas with only one type of quantifier are simpler, the interpretation of V3-formulas is more difficult for average students (an interesting point for logicians' speculations).

Furthermore, as J.P.Drouhard [92] points out, the usuai model theoretic semantics of professional mathematicians (due to Tarski) is useless to explain difficulties met by students who use algebra (even in case of simpler formulas with only one quantifier). It makes sense only for those who have already overcome difficulties we are interested in; in fact, mathematical assertions are interpreted by the language of set theory (both formally and informally): quantifiers and letters are used in the same manner as the mathematical


assertions to be interpreted. The subtlety of tarskian semantics (namely to grasp that "the snow is white" is true because the snow is white) is an end point (at a logicai level, which is stili more abstract than algebra) and not a tool to explicit difficulties in the learning of algebra.

In the following, a semantic, which seems to be more suitable for this job is developed.

3. Intensional semantics, frames and transformative functions of algebraic language

In this paragraph, two aspects of the problem are faced: namely the necessity of a precise theoric analysis of the meaning of symbolic expressions in algebra, and the use of such theoric tools to describe the dynamics of some typical algebraic processes and misconceptions.

Although ali definitions and observations are assumed to have a general value, to make clear our analysis, we shall refer to the protocol in Appendix. The problem described there is part of a set of problems posed to a sample of undergraduate students of mathematics: they have been observed while working individually and have been invited to comment aloud what they were doing.

The starting point of intensional semantics is the analysis given by logicians of semantics for modal logie (see for ex. the fundamental papers of Kripke [63a,b,c] or the recent exposition in Smorinski) enriched with some concepts from semiology and linguistics (our references in these subjects are Eco [75, 79, 84], Frege [92a,b, 77], Jakobson [56], Dummett [73], and Cauty [84]).

As observed by Drouhard, the ideas hold by Frege on semantics seem to be suitable for looking at the interpretation of symbolic expressions of algebra, insofar as its learning is concerned.

According to Frege's terminology, we shall distinguish between Sinn (sense) and Bedeutung (reference, denotation but the english translations are ambiguous) of an expression (Zeichen): the Bedeutung of an expression is the object (Gegenstand) to which the expression refers, while the Sinn is the way in which the object is given to us (everyone knows the example by Frege concerning the two different senses of Venus, namely as Esperus, the night's star, and as Phosphorus, the morning's star: the two expressions have the same denotation).

Also in Mathematics there are expressions having the same denotation but different senses. For example, the expressions 4# + 2 and 2(2x + 1) express different rules (senses) but denote the same function. Likewise, the two equations (to be solved in R) {x +1)2 = x and x2 + x + 1 = 0 denote the same object but have different senses. So mathematics, as the naturai language, has a plenty of expressions which have the same denotation.

Intensional semantics

SINN

ZEICHEN BEDEUTUNG

Frege's semi oti e tri angle

Fig. 3

The denotation of a symbolic expression in Algebra is the numerical set, possibly empty, which is represented by the expression itself. Such a set is determined both by the symbolic expression and by the numerical universe, in which the expression is considered. For ex., the above equations, while denoting the empty set in R, denote the sets { l /2 ( -9 + i\/l9), l /2( -9 - z\/l9)}, { l / 2 ( - l + i\/3), l / 2 ( - l - iy/S)}, when considered in C.

We remark that the expressions incorporate concisely in their form a sense, which we cali algebraic sense: it represents the very way by which the denoted is obtained by means of computational rules. For ex., the expression /n(n + 1)/ in the universe of naturai numbers expresses a computational rule, by which one gets the doneted set {0,2,6,12,20,...}. Algebraic transformations can produce different expressions holding different algebraic senses. For ex., transforming /n(n + l ) / into /n2 + n/ does not change the denotation but does affect its algebraic sense, i.e. the computational rule.

We should notice that it is not always true that two expressions having the same denotation can be mutually reduced by means of algebraic transformations. Conversely, algebraic transformations are not always invariant with respect to denotation (for ex., Va? ^ x).

A symbolic expression can be used in different knowledge domains, mathematica! or not, and consequently may assume additional senses depending on the nature of the domain. For ex., the expression /n{n-{-l)/ in elementary geometry may stand for the area of a rectangle of sides n, (n 4-1), whilst in elementary number theory it has the sense of "product of two consecutive numbers".

We cali contextualized sense of an expression a sense which depends on the

112 F. Arzarello - L. Bazzini - G.P. Chiappini

knowledge domain in which it lives.

Ali senses constitute the intensional aspects of an expression, while its denotation

within an universe represents its extensional aspect, which is invariant with respect to the

former ones.

As many researches have pointed out (see Drouhard [92], Kieran [89]), intensional aspects are very important because it may be very difficult for students to conceive this invariance. Very often a sort of rigidity makes students to act as if there were a one-to-one correspondence between sense and denotation. By identifying both, the former desappears and the student remains with just a trivial denotation: the symbolic expression denotes itself (usually its algebraic sense). Thus algebra becomes a pure syntax and the secondary processes (in the terminology of Sfard) are purely syntactic rules and nothing more. In these cases, the Frege's triangle (fig. 3) collapses to the segment sign-sense. The model sketched above pictures algebraic difficulties as a deficient mastering of the invariance of denotation with respect to sense. For example, pseudostructural students analyzed by Sfard do not realize, when facing equations, that "the concept of truth set - the set of numbers which must not change under the "permitted" operations - ... is where the decision to cali certain manipulations "permitted" becomes clear" (Sfard [92], p.2); since they do not see the invariance they become "formalist", in the sense that they reveal a "basic inability to link algebraic rules to the laws of arithmetic" and so "formai manipulations ... [remain] as the only source of meaning" (ibid., p.8).

This obstacle to the semantics of algebra may be rooted in the enormously easier semantics of arithmetic, which makes difficult to interpret algebraic expressions in the right way. In fact, in arithmetic the invariance of denotation is only with respect to numerical expressions whose denotation is a definite number, not a function. Let us take the point that sense and denotation are the ingredients we use to look at algebraic thinking: but they are useful only to take snapshots of algebraic difficulties and not to reconstruct the entire movie. For ex., they do not explain completely the dynamic of such processes as shown in the Appendix.

In order to understand the entire process, we will borrow some concepts from old linguistics, namely rethoric, and some others from semiology and from modal logie.

In order to support our argument, let us refer to the example sketched in the Appendix. If we look carefully to the protocol (which is a paradigmatic case), we can observe that the resolutive moment consists in the sudden change of strategy from episode 4 to episode 5. That is marked by the change of frame where the interpretative activity of the student lives1.

*We take the term frame from AI studies (for ex., see Minsky [75]); a frame is a structure of data that is able to represent a stereotyped representation, like to go to a supermarket or to a birthday party


For ex., while solving problem 1, Ann, a typical average undergraduate mathematics student, passes from the precise frame of even-odd numbers (episodes 1,2,3), to the vague frame and groping scripts of prime numbers (episodes 3,4) and then she arrives to the definite frame of multiples (episodes 4,5). The first phase is marked by stereotyped syntactical transformations guided by the even-odd numbers frame. The involution of this phase shows that the semantic control is very feeble: formulas are used as such in the frame and do not live really within it as thinking objects. Moreover, formulas do not incorporate the frame but in a shallow way: the even-odd frame is used only to test the formula, which conti nuesto be manipulated according to standard stereotypes. Anna's idea that some magic formula must be used culminates at the end of episode 3, which marks a change of frame (from even-odd to prime numbers) and a change of approach. Now Ann's reasoning becomes arithmetical; semantics is strong and the frame, which has strong numerical features, is really inhabited by numerical expressions, as thinking tools. In the long run, they activate hypothetical reasoning in a new precise frame (that of multiples), which will partially overlap the first one. At this point (episode 5) she can read the old formula with a new eye: the formula is written in such a way to incorporate the old even-odd frame and is manipulated according to an anticipative thinking related to the new frame. It is no longer the way by which the formula is written to suggest standard manipulations in order to see what happens, but, on the contrary, the formula is written and manipulated in a certain fashion in order to prove a conjecture. In other words, in the new frame, the relationship sense-denotation of the formula does appear as a thinking tool: intensional aspects are guided and built by extensional ones and conversely (episode 5, lst part). This is a first aspect of what we cali formulas as thinking tools, namely when formai manipulations are made in deep connection with denotative aspects. But there is a second way in which formulas can be used as thinking tools, as shown in the following.

The second paft of the episode shows also some stiffness of syntactical aspects: the formula incorporates the fact that 4k(k + 1) is a multiple of 8 in a transparent way only when k is even. Ann'gets some trouble when k is odd; the formai aspects force her to simplify in a certain manner and for a while the (conjectured) denotation of the formula (namely the set of even numbers) does not cope with the right algebraic sense (k + 1 is even if k is odd): to grasp this it is not required to manipulate the formula but to activate a new part of the symbolic expression according to its denotation. It is the shift in denotation (even —•> odd) which makes this possible. With respect to the first aspect of formulas as

for children. Each frame entails a certain number of informations. Some concern what one expect it is going to happen as a consequence. Others concern what one must do if such expectations are not confirmed. A frame is a sort of a condensed story, which has its scripts and where the subject is supposed to do something. Frames are activated as virtual texts while interpreting a text, for example of a problem, according to context and circumstances.


thinking tools, now the formai expression does not change; it is its sense to change, insofar as we ha ve looked at its denotation in a new way, shifting from the frame multiples to the even-odd one.

In short, the first way of thinking with a formula is transforming (part of) its intension manipulating it according to its (supposed) extension; the second way is discovering a new (supposed) intension, without doing formai manìpulatìons, but looking at a new (supposed) extension (in a possibly new jrame). In both cases such changes are activated because the intension and the extension of a formula are embedded in one or more frames.

A further comment seems important at this point. When neither purely syntactic (like in pseudoformalist students) nor syncopate aspects are overcome, average students, when facing algebraic reasoning use a doublé register, corresponding to the polarity Naturai Language - Symbolic Writing. When this amalgamation-is not perfect one of the two polarities may prevail. For ex., the first three episodes of Ann's protocol show that the even-odd frame has not been incorporated into the symbolic writing: only in the third phase a better balance is got, after the shifting of the frame. In any case, also in the latter phase, the frame multiples of 8 is incorporated with difficulty and only partially into the symbolic language. On the contrary, high students seem to use the Naturai Language code only as a framework, curtailing most of the frames used by low and average students and going directly to the last one. But, when interviewed, they have explicited spontaneously most of the curtailed frames with many details; this seems to confirm that also in these cases there may be more than one frame.

The presence of frames in solution processes gives evidence to the fact that students use a full semantics (namely with sense and denotation), which is generally marked linguistically in different ways. For ex., we have observed that Ann writes in words and stresses verbally the three successive frames.

Because of the above comments, in order to interpret properly students' algebraic processes and to explicit the meaning of algebraic thinking, it is necessary to integrate Frege semantics into a wider framework including:

(i) the use of frames;

(ii) the dynamic relationship between sense and denotation, activated by frames;

(iii) the shifting from one frame to the other.

To get this, it is necessary recalling some concept from old aged rethoric; fortunately the analysis of mathematical language from this point of view has been done by André Cauty [84]: we shall adapt his results to Frege semantics. We have seen that the main point in Frege semantics is its triangle (fig. 3); classical rethoric studies the relationship between denotation and sense from the one side (which are called collectively meaning


[lat. significatimi, fr. signifié]) and the expression which stands for them (signifying, [lat. significans, fr. signifiant]) from the other.

In particular, expressions are classified according to the fact that the same expression may be used for more than one meaning {tropos, fìg. 4) or the fact that more than one expression can be used for the same meaning (figure, fìg. 5).

We recali that typical troposes are: homonymy (/lying/ in english may concerns liars or someone in the hospital); polysemy (for ex. /penna/ in italian may concern writing (/pen/) or birds (/feather/)) and metaphor (as /the legs of a table/). The differences among troposes depend upon the relationship between the two meanings of the expression; there may be no relationship (homonymy: /lying/ refers to two completely different meanings), partial overlapping (polysemy: the two meanings of /penna/ have a partial overlapping, at least historically, when people used feathers to write), inclusion (metaphor: /leg/ usually refers to the limbs by which people stand; /leg of the table/ retains the meaning of standing but loses ali that is concerned with human beings).

Troposes rule the so called symbolic function of writing, namely when the interpretation of writing changes. In this case, the expression is not modified but its meaning changes (fìg. 4) and this usually happens because the frame has changed. The symbolic function expresses typically a semantic creativity. In fact, it implies selecting

PARADIGMATIC TRANSFORMATION (TROPOS)

1° MEANING (sense+denot.)

2° MEANING (3en3e+denot.)

Fig. 4


among different alternatives and so it is ruled by analogies and similarities. According to Jakobson's terminology, the symbolic function is a process of communication which implies the choice in a paradigm. For example, Ann choices among a paradigm for the formula, namely: even-odds, prime, multiples. In algebra, the symbolic function occurs when the frame of interpretation changes, namely the expression does not change but its meaning is modified. Generally, in algebra we are concerned with polysemies, because the frames that are picked up usually overlap. In our examples we have polysemies in episode 5, when Ann reads in another way the same formula of previous episodes (changing from the frame "even-odd" to the frame "multiples").

The second type of relationships between expressions and meanings is given by figures, namely when two different expressions have the same meaning (both denotation and sense, see flg.5). As before, there are different cases according to the fact that the meaning is exactly the same (polymorphism, like log or In: it is only the formai expression to change), or there is a partial overlapping (paraphrase or affinity of meaning, like 12 = 4*3 = 10 + 2, where denotation does not change but the sense change), or one is included in the other (methonymy, like "the function 2#" instead of "the function y = 2x").

SINTAGMATIC TRANSFORMATION (FIGURE)

MEANING sense + denotation

Fig. 5

From our point of view the affinity of meaning is the most interesting; in fact, it rules

the algorithmic function ofwrìting. By this function one can develop written computations

Intensional semantìcs 117

in arithmetic and in algebra: in fact it expresses typically syntactic effìciency. This function is only partially possessed by pseudostructural students: their transformations may lose the affìnity of meaning, because they do not own a full semantics.

The differences between the algorithmic and symbolic function of language is stressed in the following quotation by Jakobson:

"Any linguistic sign involves two modes of arrangement.

1) Combination. Any sign is made up of constituent signs and/or occurs only in combination with other signs. This means that any linguistic unit àt one and at the sante tinte serves as context far simpler units and/or finds its own context in a more complex linguistic unit.

2) Selection. A selection between alternatives implies the possibility of substituting one far the other, equivalent to the former in one respect and different from it in another. Actually, selection and substitution are two faces of the same operation.

These two operations provide each linguistic sign with two sets of interpretants...there are two references which serve to interpret the sign -one to the code, and the other to the context, whether coded or free; and in each of these ways the sign is related to another set of linguistic signs through an alternation in the former case and through an alignment in the latter.

The development of a discourse may take place along two different semantic lines: one topic may lead to another either through their similarity either through their contiguity. The metaphoric way would be the most appropriate termfor the first case and the metonymic way far the second, since they find their most condensed expression in metaphor and metonymy respectively...ln normal behaviour both processes are continually operative''' (Jakobson [56], pp.60-62).

4. Algebra as a game of interpretation and misunderstandings of students

Now our picture of algebraic thinking begins to have many ingredients and we càn use them to describe what happens when a student solves an algebraic problem, for example like that of Appendix. First she/he interprets the text of the problem; in this interpretation activity her/his cooperation is determinant to adivate one or more frames. The result is another text (may be written, but also written and spoken), which is the interpretation of the first text, and the process continues with the interpretation of this text into a new one.

For example, suppose that the problem reduces to the construction (interpretation) of


a symbolic expression E. Usually E contains some terms: unknowns, parameters or more complex expressions constructed using variables and other symbols (specific constants, operations, etc). The construction-interpretation of variables is cruciai insofar it entails two fundamental aspects: the very process of naming concepts and connotative semantics.

In introducing (or interpreting) variables (or more generally terms) one must choice in a paradigm (symbolic function): in fact, a variable contains implicitly a form of predication (for ex., p generally stands for primes or integers) and a term expresses a designation in the sense of Laborde [82] (for ex., the perimeter of a rectangle may be indicated with 2p). A very important point seems to be the rigidity that terms bave, when considered as designators. Roughly speaking, a rigid designator is a name which has only denotation and no intension (see Kripke [80]); rigid designators may be source of obstacles for algebraic reasoning.

(See in episode 5 the last diffìculties)

In addition, terms entail strong connotative aspects, in relation both to the semiotic

system used and to the contextual selection made. As a consequence, some properties

of objects denoted by the terms may be magnified or narcotized; so the process of

interpreting/constructing terms is relevant for the activation of frames and scripts (where

to develop reasonings).

For example, let us consider the following problem: "Descrive a way to get rectangle's area". Using symbolic notation, one can give the following expression: A=bh. Such an expression interprets within the symbolic language of algebra a text given in ordinary language. Other interpretations could nave been chosen, according to different semiotic systems. For ex., the following could nave been an interpretation in ordinary language: "One measures the length of the basis and of the height and multiplies them"\ or, using another semiotic system, one could bave produced a graphical interpretation. In ali cases, some properties of the object 'rectangle' and of the object 'area' are stressed and other are narcotized.

The connotative aspects are of an intensional nature, since they stress some specific

property of the denoted object. The term 'connotation' (following J.S. Mill) is sometimes

used with the same meaning as Frege 'sense'. By our opinion, 'connotation' from the point

of view of learning has a different connotation with respect to 'sense', even if both concern

intensions of symbolic expressions. To continue the joke: 'sense' and 'connotation' are

two different senses of intensional semantics: 'sense' has objective features and as such

can be communicated to other people; on the contrary, 'connotation' has typical subjective

features, related to student's experiences and culture, and its negotiation is more difficult.

Connotation concerns the aspects of so called private language (the terminology is from

Wittgenstein [53], see the discussion in Kripke [82]), which concerns the delicate point of


connection between inner experiences and officiai culture, for ex., when one grasps the rule of multiplication, or the meaning(s) of the expression /=/.

Interpretative activity is always faced with such problems; here are the roots of many misunderstandings in the learning of ali mathemàtics and of algebra in particular. In fact, mathematicians use extensional semantics in their precise definitions, but their interpretation supposes to nave already overcome misunderstandings due to intensional semantics and particularly to its connotative aspects.

Didactical engeneering of connotative semantics is the big problem of teaching algebra.

In fact, making elementary algebra means playing a game of interpretations: of a text in a semiotic system (for ex., a problem in ordinary language) into a text in another system (for ex., an equation), or from a text in a system (for ex., an algebraic expression) into a text in the same system (for ex., another algebraic expression). For ex., consider a problem to solve (e.g., a word problem, a conjecture to prove, a phenomenon to model, etc) and the request of an algebraic solution: the solution is nothing more than an interpretation and an expansipn of the text, with respect to the questions it poses. In fact, the interpretation is useful insofar as it makes possible to know something more about what is interpreted.

Fig. 6

120 E Arzarello - L. Bazzìni - G.P. Chiappini

To give a picture of algebraic thinking as a game of interpretations, let us consider again a symbolic expression E (interpreting or to be interpreted): to such an expression a set of Frege's triangles with one vertex in common (the expression) is attached: these triangles constitute ali the possible interpretations of E. When a student starts her/his interpretative activity, by various reasons she/he activates one or more frames, connected with at least one triangle. Which frame the student switches on, depends mainly on context, circumstances and connotations of the terms in the originai text. Once a frame is active, the student produces as a result of her/his interpretation a text, and the process of solution of the problem consists in successive transformations of this text, possibly in the production of completely new texts, according to the frames that are successively activated. The transformations are made according to the frames that are active in that moment and are of the two categories (tropes, figures) described by classic rethoric, namely affinities of meanings and polysemies. Of course such a distinction can be made only from an ideal point of view: in fact, in the two same types of functions there is a continuum of shadows and moreover they are deeply amalgamated in concrete students' performances.

Summarizing, the process of solution happens at two levels, namely:

(i) selection/activation/modifìcation of frames;

(ii) a) selection by analogy of formai expressions (tropes, e.g. polysemies — > symbolic function).

b) combination by contiguity of form (figures, e.g. affinity of meaning — > algorithmic function)

5. Remarks on pupils' errors according to the proposed framework

The framework described above can be adapted to interpreting some typical difficulties students encounter in algebra.

Let us look now at the two cases of bad algebraic performances, discussed at the very beginning, namely pseudostructural and syncopate strategies.

Roughly speaking, one can say that in the first case the algorithmic function is inhibited because of the supposed one-one correspondence between expression and meaning; so for a pseudostructural student, transformations are not as in figg. 4, 5 but as in fig. 7.

This anomaly can be explained with the theory of rigid designators quoted above: pseudostructural students generally use a tacit axiom, which may have a naive appeal, that each expression is a rigid designator (namely an expression with only one sense, usually the algebraic one).

As to the use of syncopated algebra , there is evidence that the symbolic function of


PSEUDOSTRUCTURAL TRANSFORMAT ION

MEANING 1 (denotati on)

MEANING 2 (denotation)

Fig. 7

algebraic language is inhibited. These students cannot have the metacognitive control (for ex., on semantics) of frames, using the symbolic language of algebra. It is like a person who cannot think in a foreign language but needs to think in her/his own language and then translate into the foreign language. In syncopate algebra, in fact, the symbolic function concerns only half of the polarities we have discussed in §1, namely naturai language and not symbolic writing (i.e. algebraic language), semantics and not syntax (of symbolic writing). Students who like syncopated algebra better are able to shift, from a Trame (and consequently from a meaning) to another only (or mainly) using ordinary language; so they do control semantically the stream of their reasoning, but only when they can use ordinary language.

It is worth noticing that the phenomenon of evaporation in the passage to symbolic writing is a typical byproduct of the above deficiencies. First it is useful clarifying the term "evaporation" by comparying it with the term condensation, according to Arzarello [92] (the word is taken from semiology, see Eco [84], p. 157, and from Freud [1905], see his analysis of the Witze). Arzarello calls condensation the process by which one constructs/grasps a tropos, namely when the same symbolic expression is referred to two different meanings, possibly in two different frames, or when one monitors the shifting


from a possible world where she/he is interpreting an expression to another one, adapting the old meanings tò the new world. Condensation stresses semantic creativity insofar as it is related to such things as analogies, metaphors, etc.

Whilst condensation entails a strong semantic control, on the contrary, evaporation concerns the dramatic loss of symbols' meaning (sense + denotation) met by some pupils when abandoning the syncopate style in algebraic problem solving. It happens that they cannot any longer express the meaning of mathematical objects in ordinary language referring to the subject's actions. Such an impossibility provokes the transition to an empty semantics (typical of pseudo-structuralist students), illustrated by fìg. 7, and to the use of symbolic expressions as rigid designators. As such, it is one of the main obstacles in the developing of an algebraic way of thinking and to a non empty use of algebraic code.

In short, condensation and evaporation both concern the symbolic functions of algebraic language, the former marking a good semantic control, the second its loss. In this delicate point, the role of language is essential, particularly as far as the dialectic between inner and outer language may bypass evaporation, provoking instead condensation. Condensation is apparently a sudden phenomenon,which happens on the spot, but it can be grasped properly if one does not look at the learning/teaching of algebra as a sequence of single acts of language but as a stream of thought, which breaks the arithmetic way of thinking.

APPENDI*

Problem. Prove that the number (p — l)(q2 — l)/8 is an even number, in case that p and q are odd primes.

PROTOCOL OF ANN (average undergraduate student of mathematics)

EPISODE 1. Ann develops the formula writing the words even, odd on the paper near the formulas:

(p - \){q2 - l ) / 8 = (p - l)(q + l)(q - l ) / 8

Ann points at the components of the formula and says: "even, even, even ... hmm ... the remaining number is not even...".

EPISODE 2. Ann develops the formula under the words even, odd:

(p- l)(q2 - l ) / 8 = (pq2-q2 -p'+ l ) / 8

makes some orai calculations of the type uodd times odd is odd, then says: ... hmm ... it does not workP1.


EPISODE 3. Like the previous one, but with calculations of the type "odd times odd is odd referred to the factors (p — 1), (q2 — 1); then Ann says : "there must be some formula for primes to use!".

EPISODE 4. Ann draws some scribbling on the formulas of preceding episodes and start verifying the formula with some primes: data are collected into a table:

P q

3 5

5 7

3 7

(5-1X25-1) 8

(5-nuq-n m 8

(3-1X49-1) 8

3

X 6

4 * dtf

6 2*4ti

fi

Ann comments: "So it is already q square minus one that is a multiple of eight!".

[between episode 4 and 5 there is no solution of continuity in the time]

EPISODE 5. Ann changes the sheet of paper and writes down the following:

8

multiple of 2 multiple of 8

Then Ann writes the following formulas:

(2h + 1 - l)[(2fc + l ) 2 - l ] /8 = 2h{4k2 + 4fc + 1 - l ) / 8 = 2hAk(k + l ) / 8

[the line of fraction is only under 4k(k + 1) and the 8 is under it]

and says: u....ifk is even, four [times] k is a multiple of8 [Ann points at the 8 in the formula], so it remains a multiple of 2 [Ann points at 2 in the formula], and we are ok. Ifk is odd ...

[Ann reduces 8 with 4 in the usuai written form, writing 2 near 8; then reduces the 2 with the 2, coefficient of h]

Ifk is odd, it does not work NO! if k is odd then k plus one [Ann points at the


k -f 1 in the formula] is even and we are overP1.

EPISODE 6. Ann looks again at the text of the problem and says: uBut primes are got nothing to do with this! Odd numbers are enough!".

The example is typical of the way average pupils reason with formulae.

High pupils instead, use at once the formula

(2h + 1 - 1>[(2A? 4- l ) 2 - l ] /8 = 2h(4k2 + 4* + 1 - l ) / 8 = 2h4k(k + l ) / 8

and argue directly as follows: "If k is even then we are over 'cause four [times] k reduces with eight; if k is odd then (k + 1) is even and the same argument applies".

Low pupils instead try generally the transformations pf episodes 1-3 or similar ones; they seem to look (more or less in a random, purely syntactical fashion) for some more complicated formula which can solve the problem and seldom try to substitute numbers for letters to see "what happens", so in the end they get lost or solve partially the problem.

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ARZARELLO F., 1991, Procedural and relational aspects of algebraic thinking, Proc. of PME XV, Assisi.

ARZARELLO F., 1992, Pre-algebraic problem solving, in: Mathematical Problem Solving Research, eds. J.P. Mendes da Ponte et al., NATO ASI Series Volumes, Springer, Berlin.

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the Learning of Mathematics, vol.9. FREGE G., 1892a, tJber Begriff und Gegenstand, Vierteljahrschrift fiir wissenschaftliche Philosophie,

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in the Leaming and teaching of Algebra, (Ed. Wagner K.), Lawrence Erlbaum Associates, National Council of Teachers of Mathematics.

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England.

Ferdinando ARZARELLO Dipartimento di Matematica, Università di Torino 10123 Torino, Italy.

Luciana BAZZINI Dipartimento di Matematica, Università di Pavia 27100 Pavia, Italy.

Giampaolo CHIAPPINI Istituto di Matematica Applicata - CNR 1-16149 Genova, Italy.

Lavoro pervenuto in redazione il 21.2.1994.

f. arzarello - l. bazzini - g.p. chiappini intensional … · 2020. 9. 20. · 110 e arzarello - l....

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