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12 Developing mastery of natural languageApproaches to some theoretical aspects of mathematics

Paolo BoeroUniversità di Genova

Nadia DouekIUFM de Nice

Pier Luigi FerrariUniversità del Piemonte Orientale, Alessandria

INTRODUCTION

Mathematical theories1 and the theoretical aspects of mathematics2 represent a challenge for mathematics educators all over the world. Neither abandoning them in favour of curricula designed to work with the average student, nor insisting on the traditional methods for teach-ing of them are good solutions. Indeed, the former represents negligence of on the part of the school in passing scientifi c culture to new generations; the latter is hardly productive and impossible in today’s school systems. This represents an interesting and important area of investigation to mathematics education research, but on what theoretical aspects of math-ematics should the effort be concentrated? Why must they be implemented in curricula? How can we ensure reasonable success with them in classroom activities?

This chapter is based on three categories of studies regarding theoretical aspects of math-ematics in school, which we and some of our Italian colleagues have undertaken in recent years: (1) studies concerning the nature of mathematical theories and theoretical aspects of mathematics in a school setting (Mariotti, Bartolini Bussi, Boero, Ferri, & Garuti, 1997; Bar-tolini Bussi, 1998; Boero, Chiappini, Pedemonte, & Robotti, 1998; Boero, Garuti, & Lemut, 1999; Boero, 2006); (2) studies based on experimental work exploring the possibility of approaching theories in primary and lower secondary school (see Bartolini Bussi, 1996; Bar-tolini Bussi, Boni, Ferri, & Garuti, 1999; Boero, Garuti, & Mariotti, 1996; Boero, Garuti, & Lemut, 2006; Boero, Pedemonte, & Robotti, 1997; Douek, 1999b; Douek & Scali, 2000; Douek & Pichat, 2003; Garuti & Boero, 2002; Garuti, Boero, & Chiappini, 1999); (3) stud-ies focused on the learning processes and the related diffi culties of Science and Engineering freshman students dealing with mathematics classes (see Ferrari 1996, 1999, 2001, 2002, 2004; Douek, 1999c)

Despite these studies’ focus on different aspects of mathematics education, mastery of natural language in its logical, refl ective, exploratory, and command functions emerges from them as one of the crucial conditions in approaching more-or-less elementary theoretical aspects of mathematics. Indeed, only if students reach a suffi cient level of familiarity with the use of natural language in the proposed mathematical activities can they perform in a satisfactory way and fully profi t from these activities. The reported teaching experiments also show how teachers must have a strong commitment to increasing students’ development of

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Developing mastery of natural language 263

linguistic competencies by way of producing, comparing, and discussing conjectures, proofs, and solutions for mathematical problems.

Theoretical positions and educational implementations concerning different functions of natural language in the teaching and learning of mathematics are widely reported in math-ematics education literature. Communication in the mathematics classroom in particular has received much attention from mathematics educators in the last two decades (Steinbring, Bartolini Bussi, & Sierpinska, 1996; Sfard, 2001; Ongstad, 2006; Saenz-Ludlow, 2006). These studies infl uenced our own research. The contributions of this chapter are intended to join the research on natural language in mathematics education through in-depth analysis of the specifi c functions that natural language plays in relationship to the theoretical side of mathematical enculturation in school and related implications for education.

Taking into account all these factors, as well as our studies and their outcomes, our chapter is organized according to a “what, why, and how” schema.

We will consider some theoretical aspects of mathematics, which are relevant both for mathematics (as a cultural inheritance) and for education in general, taking into account the needs of today’s complex society. We will describe the theoretical mathematical activity we analyse and the involved linguistic activities (our “What” question). We will provide evidence of the role of language in the development of theoretical mathematical activity, fi rst by the effects of its lack, then by its positive role at different educational levels (“Why” question). Then, (our “How” question) we will consider some conditions favouring the development of linguistic activities from the perspective of the development of mathematical activity. Some of these conditions concern early educational settings, others concern teachers’ preparation.

GENERAL THEORETICAL BACKGROUND

What theoretical aspects of mathematics?

We are interested in the mastery of specialized systems of signs (with their rules and specifi c features) in mathematical activities, as a prototype of skills that are commonly demanded in every computerized environment and in many real life situations of leisure, study or work. From this point of view, algebraic language is not an isolated case, interesting only for math-ematics, but rather one of a wide set of artifi cial languages that enter different domains of human activity.

We are also interested in the construction of mathematical objects (concepts, procedures, etc.) and their development into systems in a conscious, gradual, intentional process. Here, the focus is on transmission of mathematics as a system of “scientifi c” concepts, according to Vygotsky’s seminal work about them (see Vygotsky, 1992, chapter VI - where consciousness, intentional use and developing concepts into systems are considered crucial features of sci-entifi c concepts). Vergnaud (1990) defi ned a concept as a system consisting of three compo-nents: the reference situations, the operational invariants (in particular, theorems in action), and the symbolic representations. This defi nition can be useful in school practice because it allows teachers to follow the process of conceptualisation in the classroom by monitoring students’ development of the three components. As we will see in the next subsections, argu-mentation (i.e., the use of natural language in argumentative activities) is favourable to con-ceptualisation. Building on Vergnaud’s and Vygotsky’s elaborations about concepts, we will consider conceptualisation in the school context as the complex process that consists of the following: construction of the components of concepts considered as systems, construction of the links between different concepts, and development of consciousness about these links (Douek, 2003). The main goal will be to analyse some possible functions of argumentation in conceptualisation.


264 Paolo Boero, Nadia Douek, and Pier Luigi-Ferrari

Finally, we are interested in the construction of theories. Since the time of the Greeks, Western “rationality” has depended on peculiar forms of reasoning (e.g., deductive reason-ing), through which mathematical knowledge is organized into theories. Production of con-jectures and construction of proofs play crucial roles in the elaboration of theories. We will consider “mathematical proof” to be what, in the past and today, is recognized as such by those working in the mathematical fi eld. This approach covers Euclid’s proofs, as well as the proofs published in high school mathematics textbooks and current mathematicians’ proofs that are communicated in specialized workshops or published in mathematical journals (for the differences between these two forms of communication, see Thurston, 1994). We could try to go further and recognize some common features of proof across history—in particular, the functions of making clear and validating a statement by putting it into an appropriate frame, the reference to established knowledge (see the defi nition of theorem as “statement, proof and reference theory” in Mariotti et al., 1997), and some common requirements, like the enchaining of propositions, which, however, may differ according to the different histori-cal periods and different contexts (e.g., a junior high school context is different from that of a university graduate course context).

Today, standard presentations of mathematical theories and theorems heavily depend on specifi c formalisms. The role of formalism in mathematics, and in mathematical proofs in particular, is complex. With regard to the didactical transposition (Chevallard, 1985) of proof in school mathematics, Hanna (1989) developed a comprehensive perspective to frame fur-ther theoretical investigations and educational developments. Her paper analyses the complex interplay between the manner of presentation of mathematical results and the mathematical ideas that are to be communicated. She argues that “To a person only partially trained in mathematics [...] it might easily appear that the manner of presentation [...] is the core of mathematical practice.” This belief may induce people “... to assume that learning mathemat-ics must involve training in the ability to create this form” and then overestimating formalism, whereas “When a mathematician evaluates an idea, it is signifi cance that is sought, the purpose of the idea and its implications, not the formal adequacy of the logic in which it is couched.” The overestimation of formal aspects of mathematical proof, which has its objective reasons, also has its price because it may make students become symbol pushers. Arriving at the educa-tional implications of her analysis, Hanna argues that formalism should be regarded as a tool to be used in all its rigor when necessary (e.g., “when there is a danger that genuine confusion might develop”), but to be interpreted with some tolerance in many other situations.

In particular, this means that the use of formalism requires some metalinguistic awareness, which is more than the simple knowledge of certain rules governing formal languages (e.g., the rules of algebraic formalism) to standard, everyday-life linguistic competence.

Why language?

Bearing this perspective in mind, we focus on natural language, considered in some of its crucial functions in theoretical work in mathematics. We consider its functions:

1. As a mediator between mental processes, specifi c symbolic expressions, and logic organi-zations in mathematical activities; in particular, we consider the interplay between natu-ral language and algebraic language, and the natural language side of the mastery of connectives and quantifi ers in mathematics (see the subsections Natural and Symbolic Languages in Mathematics, and The Role of Natural Language in Advanced Mathemati-cal Problem Solving).

2. As a fl exible tool, the mastery of which can help students manage specifi c languages (command function) and which is the natural environment to develop metalinguistic awareness (in the same subsections).

3. As a mediator in the dialectic between experience, the emergence of mathematical objects and properties (i.e., concepts) and their development into embryonic theoretical systems.



(See Natural Language, Mathematical Objects, Early Development into Systems where we consider natural language primarily within individual or interpersonal argumentative activities; see also Ferrari (2003) for some examples on the role of semiotic systems in abstraction processes.);

4. As a tool in activities concerning validation of statements (fi nding counter-examples, pro-ducing and managing suitable arguments for validity, etc.; see the subsection Linguistic Skills, Argumentation and Mathematical Proof, again within individual argumentative activities).

All these functions are relevant for developing theories and theoretical aspects of math-ematics because (according to the above Vygotskian perspective about “scientifi c” concepts) theoretical work in mathematics includes, in particular, managing different systems of signs according to specifi c transformation rules (fi rst function), coherence constraints needing metalinguistic awareness (second function), connecting components of a concept as a system (Vergnaud, 1990), linking concepts into a system (third function), and deriving the validity of a statement from shared premises concerning elements of the theory and the steps of rea-soning (fourth function).

THE LANGUAGE OF MATHEMATICS: NATURAL AND SYMBOLIC

In this section, we attempt to clarify the relationships between ordinary language and the specifi c languages and notation systems of mathematics, with an emphasis on advanced math-ematics education. We argue that mathematics learning involves management of different lin-guistic varieties (registers) at the same time and that some degree of metalinguistic awareness is required to control the notation systems of mathematics, rather than specifi c profi ciency in single languages (cf. fi rst and second function of natural language as described in the previous subsection).

By language of mathematics, we mean a wide range of registers that are commonly used in doing mathematics (Pimm, 1987). According to Halliday (1985), a register is “a set of mean-ings that is appropriate to a particular function of language, together with the words and struc-tures which express these meanings.” In other words, a register is “variety according to use, in the sense that each speaker has a range of varieties and chooses between them at different times.” A thorough investigation on the evo lution of Halliday’s defi nition has been carried out by Leckie-Tarry (1995). To achieve the specifi c goal of comparing the word component and the symbolic component of mathematical language, the idea of register seems suitable in com-parison with other constructs. Advanced mathematical registers share a number of properties with literate registers, whereas to communicate in any classroom, one cannot avoid adopting everyday conversational registers. The differences between all these registers are profound, and mastery of them may require a deeper linguistic competence than the one usually displayed by students. Most mathematical registers are based on ordinary language, from which they widely borrow forms and structures, and may include a symbolic component and a visual one.

In principle, much of mathematics could be expressed in a completely formalized language (e.g., a fi rst-order language) with no word component, that is, with no component borrowed from ordinary language. Languages of this kind have been built for highly specialized pur-poses and are rarely used by people (including advanced researchers) who are doing or com-municating mathematics. We argue that ordinary language plays a major function in all the registers that are signifi cantly involved in doing, teaching, and learning mathematics.

Ordinary language in mathematical registers

Some diffi culties generally arise from the differences in meanings and functions between the word component (i.e., the words and structures taken from ordinary language) of mathemati-



cal registers and the same words and structures as are used in everyday life. The difference is not noteworthy in children’s mathematics, in which forms and meanings have been almost completely assimilated into ordinary language, but it grows more and more manifest in the transition to advanced mathematics. The needs of highly specialized languages have char-acteristics that clearly distinguish them from ordinary ones. For example, in mathematical registers, some words take on meaning that differs from the ordinary meaning, or new mean-ing is added to the standard one. This is the case of words such as power, root, or function. The use or the interpretation of some connectives may change as well, which implies that the meaning of some complex sentences (e.g., conditionals) may signifi cantly differ from the standard one. Ordinary language and mathematical language also may differ with regard to purposes, relevance, or implications of a statement. For example, in most ordinary registers a statement such as “That shape is a rectangle” implicates that it is not a square, for if it were, the word square would have been used as more appropriate for the purpose of communication. This additional information is called an implicature of the statement and is not conveyed by its content alone, but also by the fact that it has been uttered (or written) under given condi-tions. Also, a statement such as “2 is less or equal than 1000” is hardly acceptable in ordinary registers (because it is more complex than “2 is less than 1000” and conveys less information), whereas it may be quite appropriate in some reasoning processes to exploit some properties of the “less or equal” predicate. In general, a relevant source of trouble is the interpretation of verbal statements within mathematical registers according to conversational schemes (i.e., as within standard language). For more examples of this, see Ferrari (1999, 2001, 2004).

These remarks suggest not only that some degree of competence in ordinary language is required in any mathematical register, but also that working with different mathematical registers may require something more on the side of metalinguistic awareness, to manage the transition between the different conventions. The idea of metalinguistic awareness has been applied to the exploration of the interplay between language profi ciency and algebra learn-ing by MacGregor and Price (1999). In their paper, they focus on word awareness and syntax awareness as components having algebraic counterparts. In our opinion, it is necessary to consider a further component of metalinguistic awareness, namely the awareness that differ-ent registers and varieties of language have different purposes.

Ordinary language and algebraic and logical symbolisms

The relationships between the word component and the symbolic component in mathemati-cal registers are not only complex but have been developing through the years as well. For many centuries, suitable registers of ordinary language have been the main way of expressing fundamental algebraic relationships. The invention of algebraic symbolism has provided us with a powerful, appropriate tool for algebraic problems and for applying algebraic methods to other fi elds of mathematics and to other scientifi c domains (physics, economics, etc.). A widespread idea among mathematics teachers is that algebraic symbolism, once learned, is enough to treat a wide range of pure and applied algebra problems. Moreover, it is also a common belief that students’ symbolic reasoning skills develop fi rst, with the ability to solve word problems developing later. For evidence regarding this theory, see Nathan and Koed-inger (2000), who outlined the symbol precedence model (SPM), which induces a signifi cant number of teachers to fail in predicting the behaviors of two groups of high school students dealing with a set of algebra problems. We argue that all the opinions that underestimate the role of natural language in learning do not fi t the actual processes of algebraic problem solving and will give both theoretical reasons and experimental evidence.

Mathematically speaking, algebraic symbolism can be regarded as (part of) a formal system designed to fulfi ll specifi c purposes, among which we mention the opportunity of performing computations correctly and effectively.



Even if in the past algebraic symbolism was introduced as a contraction of ordinary lan-guage, and in some cases (such as “3 + 5 = 8” and “three plus fi ve equals eight”) it may be treated like that, there is plenty of evidence suggesting that the relationships between ordi-nary language and algebraic symbolism are more complex.

First of all, algebraic symbolism has a very small set of primitive predicates (in some cases, the equality predicate only); this requires the representation of almost all predicates in terms of a small set of primitives; for example, in the setting of elementary number theory, the predi-cate “x is odd” does not have a symbolic counterpart and cannot be directly translated; its symbolic representation requires a deep reorganization resulting in an expression such as “an y exists, such that x = 2y + l,” that, in addition to the equality predicate, involves a quantifi er and a new variable that does not correspond to anything mentioned in the original expression. In other words, there are plenty of algebraic expressions that are not semantically congruent (in the sense of Duval, 1991) to the verbal expressions they translate. The lack of semantic congruence may induce a number of misbehaviors such as, for example, the well-known rever-sal error (for a survey and references see Pawley & Cooper, 1997).

Another major source of trouble lies in the fact that ordinary language employs a lot of indexical expressions (such as “this number,” “Maria’s age,” “the triangle on the top,” “the number of Bob’s marbles”) that are automatically updated according to the context but are not available in algebraic symbolism. So, in a story in which at the beginning Bob has 7 mar-bles and then he wins 5 more, the expression “the number of Bob’s marbles” automatically updates its reference from 7 to 12. The same would not happen for a mathematical variable: if, at the beginning of the story, one defi nes B = the number of Bob’s marbles, then at the end, Bob has B + 5 (and not B) marbles. Some of these features of algebraic symbolism have been recognized as sources of analgebraic thinking (Bloedy-Vinner, 1996).

These peculiar characteristics of algebraic symbolism constitute its main strength (because they allow algebraic transformations, i.e., the possibility of transforming an algebraic expres-sion in such a way to both preserve its meaning and to produce a new expression that is easier to interpret or useful to suggest new meanings). But it also shows the intrinsic limitation of algebraic language in comparison with natural language; the former can fulfi ll neither a refl ective function nor a command function. In other words, the structure of symbolic expressions and the fact that symbolic translations of verbal expressions are not semantically congruent to them, even though they preserve reference or truth value, imply that they can-not be used to organize one’s processes or to refl ect on them because the nonreferential and nontruth-functional component of their meaning (e.g., connotation or the use of metaphors), which often are crucial in reasoning, are lost. Also, the use of indexicals that update their meanings according to the context is a fundamental tool for refl ection and control, as in the following sentence:

The number we have is undoubtedly divisible by three and by two because it is the prod-uct of three consecutive numbers and therefore (looking at the sequence of natural num-bers) one of them is even, one of them is a multiple of three. We could generalize this property to the product of any assigned number of consecutive natural numbers.

For further refl ection on the relationships between natural language and algebraic language from the perspective of approaching algebra in school, see Arzarello (1996).

Another case in which many teachers would prefer to make a prevalent (possibly, exclusive) use of a specialized language is the case of quantifi ers, especially in the approach to math-ematical analysis. Defi nitions and proofs are frequently written with an extensive use of the symbols for quantifi ers, up to the level of a kind of pseudo-logic formalization, as in the fol-lowing example (defi nition of continuity at xo; see Figure 12.1).



The use of quantifi ers by students is strongly encouraged as well. In this case, the neces-sity of using natural language depends on the links through natural language that we need to establish between the logic structure of a statement and its interpretation in the given con-tent fi eld. We can consider the following examples. When approaching mathematical analysis, many students consider a “decreasing” function to be the opposite of an “increasing” func-tion, and they defi ne the negation of the existence of a limit when x approaches c (“for every t > 0 a positive number d can be found, such that...”) in this way: “for no t > 0 a positive number d can be found, such that... “. The fact that “increasing” covers only one part of the functions that are not decreasing or that “for no t > 0” covers only one part of the opposition to “for every t > 0” could be a matter of symbolic transformations of formal expressions, performed according to syntactic rules. Unfortunately, in this way, novices lose all contact with meaning. Furthermore, it is not easy to move from pseudo-logic formalizations such as the one shown in Figure. 12.1 to syntactically correct expressions needed for symbolic transformations. An alternative possibility is to explain the mistake with the help of common life examples; natural language is the necessary mediator for this kind of explanation not only because the presenta-tion of common life examples requires it, but because translation from one situation to the other and related refl ective activity need natural language as a crucial tool.

Argumentation

Research about argumentation has been strongly developed over the last four decades. Dif-ferent theoretical frameworks have been proposed, based on different research perspectives (Plantin, 1990), from the analysis of the pragmatics of argumentation (argumentation to convince; Toulmin, 1958), to the analysis of the syntactic aspects of argumentative discourse (polyphony of linguistic connectives in Ducrot, 1980). A few studies have con sidered the spe-cifi c, argumentative features of mathematical activities; we may quote the cognitive analysis of argumentation versus proof by Duval (1991) and more recent studies about the interac-tive constitution of argumentation (Krummheuer, 1995) and interactive argumentation in explanation and justifi cation (Yackel, 1998). In our opin ion, if we want to consider the role of argumentation in conceptualization and proving, we need to reconsider what argumenta-tion can be in mathematical activities, focusing not only on its syntactic aspects or its func-tions in social interaction but also on its “logical” structure and use of arguments belonging to “reference knowledge” (Douek, 1999a). We use the word argumentation both for the process that produces a logically connected (but not necessarily deductive) discourse about a subject (defi ned in Webster’s dictionary as “I. the act of forming reasons, making induc-tions, drawing conclusions, and applying them to the case under discussion”), as well as the text produced by that process (Webster’s dictionary: “3. writing or speaking that argues”). The discourse context will suggest the appropriate meaning. Argument will be “a reason or reasons offered for or against a proposition, opinion or measure” (Webster’s); it may include linguistic arguments, numerical data, drawings, and, so forth. So an argumentation consists of some logically connected “arguments.” If considered from this point of view, argumentation plays crucial roles in mathematical activities: It intervenes in conjecturing and proving as a substantial component of the production processes (see Douek, 1999c); it has a crucial role in the construction of basic concepts during the development of geometric modeling activities (see Douek, 1998, 1999b; Douek & Scali, 2000).

The kind of argumentation we are interested in is as follows:

! ε > 0, ∃ δ: |× − ×0 < δ ⇒ ƒ(×) − ƒ(×0)| < ε

Figure 12.1 Current writing of the defi nition of continuity.



Production of a proposition that will be under discussion. It may be a conjecture, and it may be produced to initiate an argumentation or appear later as a result or partial result of the argumentation. A proposition may undergo a change of epistemic status through the development of the argumentation; it may turn out to be false, to be true only to a certain extent, or to be regarded as uncertain.Production of reasons to back the truth of the statement or to raise doubts about it. The reasons are taken from a reference corpus (let us say it is the shared knowledge of the class (see Douek, 2000), and they can be provided under different representations (other statements, experimental evidences, drawings, etc.). Their status may be uncertain: they depend on students’ knowledge and, more specifi cally, on students’ shared knowledge.Reasons and statements are held together by reasoning that is used to justify, to doubt, to contradict, to refute, to interpret, or to deduce new conclusions. There is a general global structure that needs to be maintained if one wants the argumen-tation to be followed and understood (one line or several converging lines). Verbal organi-zation is the perceptible aspect of such a structure.

The cognitive activity of a subject elaborating an argumentation has some characteristics too. It is a conscious and voluntary activity, it supposes the internalisation of an “other” who is in a position to control or regulate the logic of the reasoning, the truth of the statements, and the semiotic treatment of the signs involved. Exploration and trials are part of the argu-mentative activity of searching for reasons to back a hypothesis or to produce a conjecture.

SOME EXPERIMENTAL DATA AND FURTHER REFLECTIONS

Let us now examine the experimental data and specifi c refl ections concerning some of the issues discussed in the previous section. The aim will be to provide evidence of the role of lan-guage in the development of theoretical mathematical activity, fi rst by the effects of its lack, then by its positive role at different educational levels.

The role of natural language in advanced mathematical problem solving

First, we report some evidence on the role of natural language in the solution of mathematical tasks we have collected from groups of freshmen students. The choice to present data at the university level seems most signifi cant to our goals because at that level students are widely required to understand, use, and coordinate highly specialized mathematical semiotic systems, including symbolic systems such as algebraic and logical symbolism. Our evidence shows that there are ordinary language skills (related to the fi rst function of natural language—a media-tor between mental processes, specifi c symbolic expressions, and logic organizations in math-ematical activities) that are well correlated with students’ performances in algebraic problem solving. We have also found evidence that superfi cial use (i.e., with little metalinguistic aware-ness; second function) of natural language in mathematical work (according to everyday life linguistic conventions, such as conversational schemes and so on) can confl ict with specifi c semantics and conventions in mathematics, resulting in student failure.

Example 1: The fi rst experiment was carried out in 1994–95 and involved 45 fresh-man computer science students at the Università del Piemonte Orientale. The students had been offered an optional entrance test. One of the problems was a simple middle school (or lower high school) arithmetic problem (a version of the well-known “king-and-messenger problem”):

A king leaves his castle with his servants and travels at a speed of 10 km a day. At the end of the fi rst day, he sends a messenger back to the castle to be informed of the queen’s

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health. The messenger, who travels at a speed of 20 km a day, goes to the castle and departs immediately with the news. When the messenger overtakes the king, who keeps traveling at 10 km a day?

Students were asked to explain their answer, but there was no explicit mention of a written text. Nevertheless almost all of them wrote down an argument in words (sometimes with the addition of diagrams or other graphics). The papers were roughly classifi ed according to the kind of language adopted in the arguments. Three levels were identifi ed:

Level O (LO): no verbal comment, rambling words, poorly organized sentences (10 students);Level 1 (Ll): well-organized, semantically adequate, simple sentences; few compound and no conditional sentences (24 students); and Level 2 (L2): a good number of well organized, semantically adequate compound sen-tences, including conditional ones (11 students).

Thirteen students left university during October (after 2–3 weeks of university courses); among them, eight were LO, and fi ve were Ll. Twenty-fi ve students passed the algebra exam before May 1995; among them, 11 were L2, and 14 were Ll. No LO student passed the alge-bra exam before May. In other words, all students who passed the algebra exam before May were Ll or L2; all L2 students passed the algebra exam before May; and gifted students were almost equally distributed between Ll and L2.

Students in L2 seemed to have a sort of insurance against failure. Nevertheless, it was not a necessary condition for profi ciency in algebra. By comparison, the LO seems to be a suffi cient condition for failure. For more details, see Ferrari (1996). The following year, a similar exper-iment involving two tests was carried out with another group of freshman computer science students at the same university. For the fi rst test, a problem analogous to the one assigned the previous year was given, the instructions were exactly the same, and answers were classifi ed according to the same criteria. After 1 month, another problem of the same type was given, but on that occasion we asked students to write an explanation of their answers. Thirty-seven students took part in both the tests. Table 12.1 shows the outcomes.

It appears that the formulation of the task infl uenced students’ linguistic behaviors. It is noteworthy that the results of the fi rst test were well correlated with the results of the fi nal examinations (although not as well as in the 1994 test), but the results of the second test were not. The signifi cant increment of the number of students classifi ed as L2 suggests that a good share of them possess some academic linguistic skills but normally do not use them if not asked to do so. It seems that what is crucial is not so much the ability to produce high-quality texts if prompted, but the ability to use language as a tool in various situations. Notice that the fi rst case students learned (or acquired) some knowledge or skills about language as a subject, but did not use it as a semiotic tool (i.e., as a tool to use to represent ideas) to think about the problems and to communicate with others (or with oneself). In the second case, students seemed to show some form of metalinguistic awareness because they used language to represent their thought processes and were able to make choices about how to communi-cate information partly expressed in another register. These results, if confi rmed, have strong implications for the teaching of both language and mathematics.

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Table 12.1 Number of students at levels 0–2 for rests 1 and 2Level 0 Level 1 Level 2 Total

First test (October) 7 17 13 37

Second test (November) 3 7 27 37



Example 2: Let us now consider a confl ict between ordinary language and mathematical language. Situations of this kind may occur often, especially at the advanced level. In the fol-lowing problem, we report, as an example, the case of two terms designating the same referent (an unusual situation in everyday-life registers):

Problem: Is it true that the set A = (-1, 0, 1) is a subgroup of (Z, +)?

There are students who claim that A is closed under addition and show that if an element of A is added to another (different) element of A, the result belongs to A. They do not take into account the case x = y = 1 nor x = y = -1, the only evaluations that lead to discover that A is not closed under sum (i.e., that sum is not a function from A x A to A). In other words, they mis-interpret the defi nition of subgroup. This happens despite students’ knowledge of different representations (such as addition tables) that point out that an element can be added to itself. Moreover, if explicitly prompted, they seem aware that each of the variables x, y may assume any value in A. Most likely, they are hindered by the need to use two different variables to denote the same number, which does not comply with the conventions of ordinary language according to which different expressions (in particular, atomic ones) usually denote different things. Notice that phenomena of this kind involve the pragmatic dimension of languages because they concern language use more than the simpe interpretation of symbols (students seem aware that each of the symbols x, y may denote any element of A but nonetheless use them to denote pairs of different elements only).

Some comments

The experimental data we have presented can be interpreted in a unifi ed way if we accept an analysis of the “language of mathematics” that takes into account the role of natural lan-guages and avoid overvaluing the role of specifi c symbolism.

First, we remarked that students do not use languages fl exibly; they often do not apply their linguistic skills to the resolution of problems (Example 1) or apply conversational schemes improperly (Example 2). Thus, a fi rst goal for mathematics education that comes out from our evidence is the need to teach not only languages (from ordinary to specifi c symbolic ones), but also the fl exible use of them. In this perspective, ordinary language (which is far more fl exible than specifi c mathematical symbolisms) should play a major role.

Second, we remarked that students often lose their contact with meaning. As shown in Example 2, the meaning of the defi nition of subgroup (as far as it could be grasped through alternative representations or students’ knowledge or their previous mathematical experience) is neglected and a stereotyped interpretation is adopted. In this regard, ordinary language, as a refl ective and command tool in the interplay between semantic and syntactic aspects of algebraic and logic activities, could help students in keeping themselves in touch with mean-ings. Of course, ordinary language alone cannot guarantee meaningful learning, but it can act as mediator between everyday experience and the specifi c needs of mathematical thinking, in particular, the need to interpret and apply patterns of reasoning that are different from the customary ones. Natural language is designed to represent a wide range of everyday-life meanings and patterns of reasoning and embodies most of them, but it is also a fl exible tool that can be used to express different meanings (e.g., truth-functional semantics) and different logics (e.g., mathematical logic). Thus, a fl exible mastery of ordinary language (which cannot be achieved by means of everyday-life experiences alone but should include scientifi c commu-nication) should be a necessary step to mathematical profi ciency.

The available data suggest one main educational implication: the need for developing mas-tery of natural language in mathematical activities as the key for accessing control of algebraic problem-solving processes. These data stress also the necessity of considering opportunities



and limitations of that particular, specialized “mathematical verbal language” that includes mathematical symbols, peculiar and often stereotyped mathematical expressions (“for every t > 0 a positive number d can be found, such that…, etc.”) as a mediator between the fl exibility of ordinary language and the specifi c needs of mathematical activities.

Natural language, mathematical objects, early development into systems

Recent literature has widely considered the issue of the constitution of mathematical knowl-edge through language, according to different theoretical orientations; in particular, for rea-sons of proximity with our own research, we may quote Sfard’s constitution of mathematical objects through linguistic processes (Sfard, 1997, 2000). In this section, we consider a spe-cifi c issue: the role of natural language in the constitution of mathematics concepts through argumentation (i.e., we consider some aspects of the third function of natural language as a mediator in the dialectic between experience, the emergence of mathematical objects and properties, and their development into embryonic theoretical systems).

Developing reference situations: Experiences, reference situations and argumentation

An experience can be considered as a reference situation for a given concept when it is referred to as an argument to explain, justify, or contrast in an argumentation concerning that concept. The criterion applies both to basic experiences related to elementary concept construction and to high-level, formal, and abstract experiences. As a consequence of our criterion, to become a reference situation for a given concept, an experience must be connected to symbolic rep-resentations of that concept in a conscious way (to become an argument intentionally used in an argumentation). In this way, a necessary functional link must be established between the constitution of reference experiences for a given concept and its symbolic representations. Argu mentation may be the way by which this link is established (see Douek, 1998, 1999b and Douek & Scali, 2000 for examples and Bernstein, 1996, for a general perspective about recontextualization of knowledge).

Argumentation and operational invariants

Argumentation allows students to make explicit operational invariants and ensure their con-scious use. This function of argumentation strongly depends on teacher’s mediation and is fulfi lled when students are asked to describe and discuss effi cient procedures and the condi-tions of their appropriate use in problem solving. The comparison between alternative pro-cedures to solve a given problem can be an important manner of developing consciousness. The inner nature of concept as a “system” is enhanced through these argumentative activities: operational invariants can be compared and connected with each other and with appropriate symbolic representations, thus revealing important aspects of the system.

Argumentation, discrimination, and linking of concepts

Argumentation can ensure both the necessary discrimination of concepts and systemic links between them. These two functions are dialectically connected: Argumentation allows us to separate operational invariants and symbolic representations between similar concepts (for an example, see Douek, 1998, 1999b: Argumentation about the expression height of the Sun allows one to distinguish between height to be measured with a ruler and angular height of the Sun to be measured with a protractor). In the same way, possible links between similar concepts can be established.



An example

The example comes from a primary school class of 20 students, a participant in the Genoa Project for Primary School. The aim of this project is to teach mathematics, as well as other important subjects (native language, natural sciences, history, etc.), through systematic activi-ties concerning “fi elds of experience” from everyday life (Boero et al., 1995). For instance, in fi rst grade, the “money” and “class history” fi elds of experience ground the development of numerical knowledge and initiate argumentation skills, as well as the use of specifi c symbolic representations.

We consider data coming from direct observation, the students’ texts, and videos of class-room discussions.

Grade II: Measuring the height of plants in a pot with a ruler

We analyze a classroom sequence consisting of fi ve activities concerning the same problem: Students had to measure with a ruler the height of wheat plants grown in the classroom. The students had previously measured the wheat plants when taken out of the ground and were to determine the increase in heights of plants over time. The diffi culty was in the fact that rulers usually do not have the zero mark at the edge, and students were not allowed to push the ruler into the ground (to avoid harming the roots). The children had to fi nd a general solution (not one that worked for a specifi c plant). In particular, they could use either the idea of translating the numbers written on the rulers (by using the invariability of measure through translation), an act we call the translation solution, or the idea of reading the number at the top of the plant and then adding to that number the measure of the length between the edge of the ruler and the zero mark (by using the additivity of measures), an act we call the additive solution.

The fi rst activity was a one-on-one discussion with the teacher to fi nd out how to measure the plants in the pot. The ruler had a 1 cm space between the zero mark and its edge. The purpose of this discussion was for the students to arrive at and describe solutions. The main diffi culties met by students were as follows.

First, students found it diffi cult to focus on the problem. The teacher, T, used argumenta-tion (in discussion with the student, S) to focus on the problem, as in the following excerpt: (with the help of the teacher, this student had already discovered that the number on the ruler that corresponds to the edge of the plant is not the measure of the plant’s height):

S: We could pull the plant out of the ground, as we did with the plants in the fi eld.T: This would not allow us to measure the growth of our plants.S: We could put the ruler into the ground to bring zero to the ground level.T: But if you put the ruler into the ground, you might harm the roots.S: I could break the ruler, removing the piece under zero.T: It is not easy to break the ruler exactly on the zero mark, and then the ruler would

be damaged.

Once focused on the problem, the question remained of how to go beyond the knowledge that the measure line on the ruler was not the height of the plant. Usual classroom practices on the “line of numbers” (shifting numbers or displacing them by addition) had to be trans-ferred to a different situation. A change of the ruler’s status was needed: Instead of a tool for measurement it became an object to be measured or transformed (e.g., cut or bent) by the imagination. With the help of the teacher, most students overcame diffi culties using different methods. In particular, some of them imagined putting the ruler into the ground to bring the zero mark to the level of the ground, but because this action could harm the roots, they imagined shifting the number scale along the ruler.



S: I could put the ruler into the ground ...T: If you put the ruler into the ground, you could harm the roots.S: I must keep the ruler above ground ... but then I can imagine bringing zero below,

on the ground, and then bring one to zero, and two to one ... It is like the numbers slide downward.

Other students imagined cutting the ruler. They were not allowed to do so, so they imagined taking a piece of the ruler (or the plant) and bringing it to the top of the plant.

At the end of the interaction, 9 students out of 20 arrived at a complete solution (i.e., they were able to dictate an appropriate procedure): Four were translation solutions, four were additive solutions, and one was a mixed solution (with an explicit indication of the two pos-sibilities, addition and translation).

Rita’s translation solution: To measure the plant we could imagine that the numbers slide along the ruler, that is, 0 goes to the edge, 1 goes where 0 was, 2 goes where 1 was, and so on. When I read the measure of the plant, I must remember that the numbers have moved: If the ruler gives 20 cm, I must think about the number coming after 20 - 21.

Alessia’s additive solution: We put the ruler where the plant is and read the number on the ruler, which corresponds to the height of the plant, and then add a small piece, that is, the piece between the edge of the ruler and 0. But, we must fi rst measure that piece.

Four students moved toward a translation solution without being able to make it explicit at the end of the interaction. The other seven students reached only the understanding that the measure read on the ruler was the measure of one part of the plant and that there was a “missing part,” without being able to establish how to go on.

The second activity consisted of an individual written production. The teacher presented a photocopy of Rita’s and Alessia’s solutions, asking the students to say whose solution was like theirs and why (inviting students to produce elementary argumentation to relate their solu-tion to those of others). This task was intended to provide all students with an idea about the solutions produced in the classroom, and to allow them to build links between their solutions, their knowledge, and the procedures contextualised and described in natural language. With one exception, all the 13 students who had produced or approached a solution were able to recognize their solution or the kind of reasoning they had started; thus, they could surpass the particularities of the various linguistic or representational elaborations. Six out of the other seven students declared that their reasoning was different from that produced by Rita and Alessia.

The third activity was a classroom discussion. The teacher worked at the blackboard orches-trating a collective argumentation, and the students worked on their exercise books (where they had drawn a pot with a plant in it). They used a paper ruler similar to the teacher’s, effec-tively putting into practice the two proposed solutions, fi rst the “translation” solution and then the “additive” solution. The “arrow” representation of addition was spontaneously used by some students to represent either the shifting of the number scale along the ruler, or the transfer of the bottom piece of the ruler to the top of the plant. This involved simultaneous extension and linking of the semiotic tools used. Meanwhile, students discussed some prob-lematic points that emerged. In particular, they noted that while the “translation” procedure was easy to perform only in the case of a length (between the zero mark and the edge of the ruler) of 1 cm (or eventually 2 cm), the “additive” solution consisted of a method easy to use in every case. Another issue they discussed concerned interpretation of the equivalence of the results provided by the two solutions (“Why do we get the same results?”).

The fourth activity was an individual written production in which students had to explain why Rita’s method works, and explain why Alessia’s method works. By this point, they had



developed or been in touch with varied arguments that could be reorganised in their personal argumentative text, and we will see hereafter that the students do assimilate these arguments and produce a coherent personal text, which also proves that they assimilated the mathemati-cal procedure, even if the mathematical representation remains absent.

With three exceptions (who remained rather far from a clear presentation, although they showed an “operational” mastery of the procedures), all the students were able to produce the explanations the task demanded. Half of the students added comments about the two methods; most of them explained in clear terms the limitation of the “translation” solution. Here is an example:

Marco: Rita’s reasoning works, because it makes the ruler like a tape measure, because zero goes at the edge of the ruler. She imagined the same thing and made the numbers slide. Where I see that the edge of the plant is at 8 cm, she says “to slide” and sees that the ruler slid by one centimeter, and so she sees that 8 became 9. But this method works easily only if the ruler has a space of 1 cm between zero and the edge. Alessia’s reasoning works because she adds the piece of plant she carries to the height of the plant and adds 1 to 10 and she sees that it makes 11. One differ-ence is that Rita leaves the piece of the plant where it is, while Alessia carries it up to 10.

The fi fth activity consisted of the classroom construction (guided by the teacher) of a syn-thesis to be copied into students’ exercise books.

In the following classroom activities, students recalled “measure of the plants in the pot with the ruler” as a reference when they had to measure the length of objects that were not accessible in a direct reading of their length on the ruler (for further details, see Douek, 2003; Douek & Scali, 2000).

Some comments on the available data

The available data show experimental evidence for the potential of natural language involved in argumentative activities, as stated previously. In particular, during the interactive resolu-tion of the problem (the fi rst activity), the argumentative activity with the teacher allowed students to grasp the nature of the problem and transformed the experience into a possible “reference situation” for the involved operational invariants of the measure concept. The teacher’s arguments compelled students to move from imagining physical actions (putting the ruler into the ground or cutting the ruler) to operations involving operational invariants of the measure concept (translation invariance or additivity of measure of length).

During the classroom discussion (the third activity), argumentation allowed students to make explicit the two different operational invariants (translation invariance and additivity) of the measure of lengths and the systemic links between them and with other concepts (addi-tion and subtraction) in the conceptual fi eld of the additive structures (Vergnaud, 1990).

During the last individual activity (“explain why...”: a typical task requiring argumenta-tion), students attained a fi rst level of understanding about the potential and limitations of the operational invariants involved.

LINGUISTIC SKILLS, ARGUMENTATION, AND MATHEMATICAL PROOF

In this section, we will refer to the second and fourth functions of natural language as described in the second section of this chapter. In particular, we will try to support the idea that rich argumentative processes (including management of analogies, metaphors and ques-tioning) constitute the core of the activities of conjecturing and proving, while sophisticated



metalinguistic awareness and linguistic competences are needed to obtain socially acceptable products (“proofs”) (Boero, 2006). In order to achieve our goal, we need to discuss the relationships (and the distinction) between proof as a product (submitted to social rules of conformity to cultural models) and proving as a process, and also to take a critical position about the widespread conviction (shared by many mathematics teachers and some mathemat-ics educators) that learning to prove mainly concerns the development of formal logic skills. In particular, the image of proving that we want to highlight is that of a complex, culturally rooted but also creative practice needing a highly developed mastery of natural language in its refl ective, control and command functions. An example from university mathematics students will provide some evidence for our position.

The questions related to formalism are relevant to the debate on argumentation and math-ematical proof in mathematics education, as far as formalism is often regarded as a sort of guarantee for the soundness of mathematics results, and may become a serious obstacle to the development of mathematical reasoning.

Mathematical proof and argumentation as linguistic products

To compare mathematical proof and argumentation as linguistic products, we adopt the fol-lowing criteria of comparison, inspired in part by Duval’s analysis (see Duval, 1991): the role of context (and, in particular, the existence of a “reference corpus”) in the development of reasoning; and the form of reasoning.

Two of the most relevant points of discrimination between argumentation and mathemati-cal proof as products are the language they adopt and the context they work within. The con-text of argumentation includes all the levels of context, as identifi ed by Leckie-Tarry (1995) in the frame of a functional perspective, that is, the context of situation (the concrete, physical, and social context the exchange taking place within), the context of text (the information pro-vided by the texts involved, e.g., the text of the problem under scrutiny and other connected texts), and the context of culture. The context (and consequently the reference corpus) of mathematical proof is quite different: The context of situation is usually banned, and even the context of culture is strongly bounded. Not all the knowledge available can be used, but only parts of it according to respective operational status. For example, when writing the proof of some propositions from Euclid’s Elements, one could use any proposition that has already been proven (i.e., one occurring before in the list of propositions) but no other; more gener-ally, a mathematical proof often refers to “straightforward” arguments, but not all arguments that may appear straightforward to a non-expert (e.g., measure, for plane geometry fi gures) are allowed. Also from the viewpoint of “language genre,” argumentation and proof (as prod-ucts) are different. Argumentation may use a wide range of linguistic registers, including con-versational (and situation-dependent) ones, whereas mathematical proof, for various reasons, is compelled to use more explicit and institutionalized ones only. In particular, argumentation may freely use a wide range of metaphors, whereas in mathematical proofs only few metaphors are allowed (e.g., large-small, before-after) according to conformity criteria. According to Leckie-Tarry’s classifi cation of contexts and registers, this means that the understanding and production of mathematical proofs belong to the literate side of linguistic performances and require prior deep linguistic competence. In the preceding comments, we stressed relevant points of the difference between argumentation and mathematical proof as products. Now, we point out some aspects of at least partial superposition. If we focus on the part of context Duval named “reference corpus,” which includes not only reference statements but also visual and, more generally, experimental evidence, physical constraints, and so forth assumed to be unquestionable (i.e., “reference arguments,” or, briefl y, “references,” in general), we under-stand that no argumentation (individual or between more participants) would be possible in everyday life if there were no reference corpus to support the steps of reasoning. The refer-ence corpus for everyday argumentation is socially and historically determined. Mathematical



proof also needs a “reference corpus,” which is socially and historically determined as well. In addition, the reference corpus is generally larger than the set of explicit references, both in mathematics and in other fi elds. In mathematics, the knowledge used as reference is not always recognized explicitly (and thus appears in no statement); some references can be used and might be discovered, constructed, reconstructed, and stated afterward. The example of Euler’s theorem discussed by Lakatos (1976) provides evidence about this phenomenon in the history of mathematics. The same phenomenon also occurs for argumentation concern-ing areas of study other than mathematics. In general, we could hold no exchange of ideas, whatever area we are interested in, without exploiting implicit shared knowledge. Implicit knowledge, of which we are generally unconscious, is a source of important “limit problems” (especially in non-mathematical fi elds, but also in mathematics). In the “fuzzy” border of implicit knowledge, we can meet the challenge of formulating more and more precise state-ments and evaluating their epistemic value. Again Lakatos (1976) provided us with interesting historical examples about this issue in mathematics. Another crucial point of superposition between argumentation and mathematical proof is the relevance of “content” (semantic argu-ments) in the validation of statements. This fact contrasts a widespread idea in the community of mathematics educators, according to which advanced mathematical proofs are near to the model of formal derivation, and checking their validity is a formal logic exercise. In the reality of today’s university mathematics, if we consider some basic theorems in mathematical analy-sis (e.g., Rolle’s theorem, Bolzano-Weierstrass’ theorem, etc.), we see that their usual proofs in current textbooks are formally incomplete (if considered as formal derivations), and com-pletion would bring students far from understanding; for this reason semantic (and visual) arguments frequently are exploited or evoked to fi ll in the gaps existing at the formal level. We add that the validity of most of published mathematical proofs is based on the semantics of symbolic (linguistic, algebraic, etc.) expressions that constitute them and that checking for validity is based on strategies that usually are very different from checking the validity of a formal derivation. As Thurston (1994) noted,

Our system is quite good at producing reliable theorems that can be solidly backed up. It’s just that the reliability does not primarily come from mathematicians formally check-ing formal arguments; it comes from mathematicians thinking carefully and critically about mathematical ideas.

The processes of argumentation and construction of proof

We have proposed distinguishing between the process of construction of proof (proving) and the product (proof). What comprises the “proving” process? Experimental evidence has been provided about the hypothesis that in many cases “proving” a conjecture entails establishing a functional link with the argumentative activity needed to understand (or produce) the state-ment and recognizing its plausibility (see Mariotti et al., 1997). “Proving” requires an inten-sive argumentative activity, based on “transformations” of the situation represented by the statement. Experimental evidence about the importance of “transformational reasoning” in proving has been provided by various studies (see Arzarello, Micheletti, Olivero, & Robutti) 1998; Boero et al., 1996, 1999; Harel & Sowder, 1998; Simon, 1996). Simon defi ned trans-formational reasoning as follows:

the physical or mental enactment of an operation or set of operations on an object or set of objects that allows one to envision the transformations that these objects undergo and the set of results of these operations. Central to transformational reasoning is the ability to consider, not a static state, but a dynamic process by which a new state or a continuum of states are generated.



It is interesting to compare Simon’s defi nition with that of Thurston (1994):

People have amazing facilities for sensing something without knowing where it comes from (intuition), for sensing that some phenomenon or situation or object is like some-thing else (association); and for building and testing connections and comparisons, hold-ing two things in mind at the same time (metaphor).

Metaphors can be considered as particular linguistic outcomes of transformational reasoning. For a metaphor, we may consider two poles (a known object and an object to be known) and a link between them. In some cases, the “creativity” of transformational reasoning consists of the choice of the known object and the link, which allows us to know some aspects of the unknown object as suggested by the knowledge of the known object (termed “abduction” by Arzarello et al., 1998). We note that mathematics “offi cially” concerns only mathematical objects. Usually, metaphors where the known pole is not mathematical are not acknowledged. But in many cases, the process of proving needs these metaphors, with physical or even bodily referents (Thurston, 1994). In general, Lakoff and Nunez (1997) suggested that these meta-phors have a crucial role in the historical and personal development of mathematical knowl-edge (“grounding metaphors”).

Two examples from university mathematics students

Forty-three fourth-year university mathematics students had to generalize a proposition (the sum of two consecutive odd numbers is divisible by four) then prove the generalized proposi-tion. Students attended a mathematics education course and were accustomed to reporting their reasoning verbally. Here are two examples of students’ texts. They represent characteris-tic and opposite behaviors in the texts of many students.

Example 1

Excerpts from the text of Student 1 contain seven large, spatially organized pieces, such as the two reported below; and many arrows, connecting lines, and encirclings.

I have some diffi culties understanding in what direction I must generalize. It might be: “by adding two odd or even consecutive numbers I get a number divisible by 4”

[She performs some numerical trials]This does not work. I will try to generalize in another way (see Figure 12.2 and 12.3)

Figure 12.2 An excerpt from the protocol of Student 1: Even numbers.

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I was looking for something that could help me… But I have nothing.

[Other trials, with a rich spatial organization; including two consecutive even num bers, two consecutive odd numbers; here she gets divisibility by 4; then three, four, fi ve, six, and seven consecutive odd numbers. By performing calculations, she gets the following formulas:

3(2K + 3);8(K + 2);

10K + 25 = 5(2K + 5);12K + 36 = 12(K + 3);

14K + 49 = 7(2K + 7) ].

Is the result of the addition of n consecutive odd numbers (n odd) divisible by n?(2K+ 1) + (2K + 3) + --- (2K+)? What must I put here? (E)

Figure 12.3 An excerpt from the protocol of Student 1: divisibility by n.

[The student performs an unsuccessful trial by induction, then considers n numbers in general.]

n numbers: (2K + 1) + (2K + 3) + … (2K + (2n - 1)) = 2nK + 1 + 3 + 5 +…+ (2n - 1) = 2nK + …(I am thinking of the anecdote of “young Gauss”: (F) it makes 2n . n/2 = n2) = 2nK+ n2 = n(2K+ n) OK!!

[Trials performed by applying the preceding formula 2nK + n2 in the cases n=2, n = 4, n = 6, n = 8: The student arrives at: 4K + 4 divisible by 4;8 K + 16 divisible by 8;12K + 36 = 12(K + 3) divisible by 12;16K + 64 = 16(K + 4) divisible by 16].

Then if I add n consecutive odd numbers (n even), I get divisibility by 2n. Let us try a proof:(P) (2K + 1) + (2K + 3) + ...(2K + 2n - 1) = 2nK + (1 + 3 + ... + 2n - 1) = 2nK +(2n.n)/2 = 2nK+ n2 = 2n(K + n/2).n even implies that n/2 is an integer number: so I get divisibility by 2n.

Example 2

In the excerpts from the text of Student 2, spatial organization is almost linear, like that in the following transcript.



Student 2 starts her work by checking (on numerical examples: 3 + 5; 5 + 7; 101 + 103) the validity of the given property and then proves it. Then she writes:

What does it mean “to generalize”? It means considering a property in which there are some closed variables (two odd numbers, or divisibility by 4) and getting a property in which variables are open. I change the number of odd consecutive numbers to add.

For instance, I consider 3 [crossed out] 4 consecutive odd numbers 2n + 1, 2n + 3, 2n + 5, 2n + 7 and make the addition: 2n+l +2n+3+2n+5+2n+7=8n+16 = 8(n + 2) = 4(2n + 4). Then I fi nd a number that is divisible by 8, so it is divisible by 4. I perform the addition of six consecutive odd numbers [similar calculations] = 12n + 36 = 6(2n + 6). Then I fi nd a number divisible by 12, so it is divisible by 4. 1 try with 8: [similar calculations] = 8 . 2n + 64 = 8(2n + 8). Then I fi nd a number that is divisible by 8, so it is divisible by 4. Following my reasoning, for an even number K of odd consecutivenumbers, I get: 2n + 1 + 2n + 3 + ... + 2n + 15 + … = K(2n + K) = 2K(n + K/2); but K is an even number, so it is divisible by 2 and (n + K/2) is an integer number. Then 2K is divisible by 4 (because K is odd). So I have found that the given property is still valid if I add up an even number of odd consecutive numbers.

Analysis of student behaviors

We attempt to show the complexity of the argumentative activity needed to fulfi ll the task; such complexity involves different functions of natural language in interaction with other symbolic systems.

The task called for elementary content reference knowledge: elementary arithmetic, alge-braic language, and its rules of calculation. Concerning algebra, we may

remark that the process of formalization (i.e., the passage from content to formula) was not easy for many students, especially when they wanted to write the sum of K odd. For instance, some of them wrote (2n + 1) + (2n + 3) + … + (2n + ?) and then stopped; few were able to express ? as 2K - 1: see (E) in Example 1.

Writing the result of the sum was not easy. It demanded a semantically rooted conversion of a known formula (the formula for the sum of the fi rst n natural numbers) or the reconstruc-tion of an ad hoc formula: see (F) in Example 1.

In general, natural language was exploited to refl ect on the situation and manage the for-malization and interpretation processes.

With regard to the external representation of content reference knowledge, we have found many organizations of data and schemas with visual effects that reveal regularities and help to express algebraically some arithmetic relations; we also found symmetries in the disposition of data and formulas, which provide hints for the calculus (see Example 1 for two cases of this). Natural language was primarily used to interpret what emerged from the external nonverbal representations.

Metamathematical knowledge

Summing up the analyses performed, we may say of metamathematical knowledge that shared explicit knowledge was much narrower than the actual knowledge used globally by the stu-dents. We found that more than half of students referred explicitly to methods for solving problems of this kind, but, as an example, “organization of data” was never mentioned even in partial explications of methods although it was a key strategy for 12 students and useful for nine of them. The implicit problem-solving methods we could detect globally were change of representation, interpreting calculations in words and vice versa, and visually organizing data and calculations. We also detected many changes of mathematical frames: arithmetic, algebra, series, and so forth.



Natural language played a crucial role in the management of metamathematical knowl-edge; in particular, it accomplished command and refl ective functions about changes of frame, changes of representation, and other factors.

Algebraic-syntactic or semantically based steps of reasoning

We have listed numerous breaks during calculations, which were needed to reinterpret the mathematical content of calculus in words. Wording was a crucial tool for reinterpretation. This can be seen as a sign of the primacy of semantic content over algebraic calculation during the process of conjecture and proof construction. As an example, we can consider the need of Student 1 to express algebraic propositions in words when seeking to recognize possible con-jectures. This attitude displays the search for a semantically consistent grasp of the algebraic signs. We can interpret it by saying that constructive work in mathematics cannot evolve within formal expression alone. Natural language is revealed here as a crucial tool for thinking.

Proof as product and proof as process

As previously stated, analyzing the text of Student 1 and other students who performed well, we can observe frequent changes of strategy, organization of data, and calculations, as well as a frequent effort to interpret the problem with words. Some of these useful forms disappeared in the fi nal drafts of the proof, in which the logical link between the propositions became a priority (see P in Example 1). Also, justifi cation of the research method disappeared from the products, whereas examples of the interwoven presence of metamathematical arguments in mathematical reasoning were frequent in the construction stage.

Student 2 is considered skillful; her presentation is close to that of a fi nal presentation. Nonetheless, this approximation to a formally correct mathematical text (Hanna, 1989) seems to bear negative consequences on the productivity of this student’s work: Her research is linear, and no change of strategy is found at any level. There are long repetitive arithmetic calculations, quite surprising for the only student in the group who usually managed alge-braic tools very well; more remarkably, the student arrived algebraically at a strong conjecture and interpreted it in words as being much weaker. Finally, she did not produce a complete proof. The same happened with other students producing texts that were similar to a fi nal presentation.

Conclusions

We have seen that important reference knowledge remained implicit in the students’ proving processes and that some of the references concerned the content, whereas others related to the metaknowledge about the activity to be performed. Natural language played a crucial role in the management of the complex game involving explicit as well as implicit reference knowledge. We also have seen how nonstandardized, appropriate representation of explicit reference knowledge played an important role in the students’ performances, under control of natural language. We have seen that when elaborating a productive process, many students found syntactic arguments insuffi cient, and so semantically rooted arguments (expressed in words) became critical. Finally, we have collected some experimental evidence about the nega-tive consequences of subordinating the proving process to the requirements of proof as a fi nal product, as a “standardized text.”

Some educational implications

Let us come back to the argumentative process of proof construction as distinct from the fi nal result. In our opinion, an important part of the diffi culties of proof in school mathematics



comes from the confusion of proof as a process and proof as a product, which results in an authoritarian approach to both activities. Frequently, mathematics teaching is based on the presentation (by the teacher and then by the student when asked to repeat defi nitions and theorems) of mathematical knowledge as a more or less formalized theory based on rigorous proofs. In this case, authority is exercised through the form of the presentation (see Hanna, 1989); in this way school imposes the form of the presentation over the thought, leads to the identifi cation between them, and demands a thinking process modeled by the form of the presentation (eliminating every “dynamism”). This analysis may explain the strength of the model of proof, which gives value to the idea of the linearity of mathematical thought as a necessity and a peculiar aspect of mathematics. The use of natural language is molded to such constraints of linearity, losing its potential for creativity (e.g., producing and using metaphors, managing explorations, etc.).

If a student (or a teacher) assumes linearity as the model of mathematicians’ thinking with-out taking the complexity of conjecturing and proving processes into account (consider the example of Student 1), it is natural to see “proof” and “argumentation” as extremely differ-ent. On the contrary, giving importance to nondeductive aspects of argumentation required in constructive mathematical activities (including proving) can develop different potentials. Regarding the possibility of educating ways of thinking other than deduction, Simon (1996) considered “transformational reasoning” and hypothesized:

transformational reasoning is a natural inclination of the human learner who seeks to understand and to validate mathematical ideas. The inclination ... must be nurtured and developed.... school mathematics has failed to encourage or develop transformational reasoning, causing the inclination to reason transformationally to be expressed less universally.

We are convinced that Simon’s assumption is a valid working hypothesis, needing further investigation not only regarding the role of transformational reasoning in classroom discourse aimed at validation of mathematical ideas but also its functioning and its connections with other “creative” behaviors (in mathematics and in other fi elds).

In this way, argumentation seems an activity suitable to promote both the improvement of linguistic skills (e.g., forcing the transition from conversational, oral registers to more abstract, written ones) and the development of mathematical reasoning. In particular, through argu-mentation in social contexts and the activity of writing down reports of the related discus-sions, experiential knowledge progressively becomes textual, and some of the arguments that were implicit in the context of the social situation become explicit in the context of individual texts and then, in the context of the shared culture of the class.

The passage from argumentation to proof about the validity of a mathematical statement should be constructed openly because of limitation in the reference corpus (see Mathemati-cal Proof and Argumentation as Linguistic Products). This passage could be supported by using different texts, such as historical scientifi c and mathematical texts, and different modern mathematical proofs (see Boero et al., 1997, for a possible methodology).

HOW CAN STUDENTS DEVELOP NATURAL LANGUAGE COMPETENCIES NEEDED IN MATHEMATICAL ACTIVITIES?

Research perspectives and problems about the role of natural language in mathematical activi-ties considered in the preceding sections raise important questions related to teachers’ prepa-ration and educational implications for classroom work. We now consider some aspects of this problematique and some didactical implications.



Approaching written language and the early development of mathematical argumentation

The aim now is to elaborate (and provide some experimental evidence for) the hypothesis that, during the fi rst-grade students’ approach to writing, an appropriate educational setting based on social interaction and involving students in some shared experience can give them the opportunity to develop important skills related to mathematical argumentation. In particular, this can be an early step to enabling students to prove. The role of classroom discussions and one-on-one teacher-student interactions, and the importance of choosing suitable objects for interactions, will be considered.

In the last decade, the early development of students’ argumentative skills progressively became a subject of major concern for mathematics educators for different reasons: the need for an early approach to skills that are relevant in the proving process (under the pressure of curricular changes that led to a re-evaluation of mathematical proof in school; see NCTM Standards, 2000); the potential of classroom interaction in developing mathematical knowl-edge and skills (see Krummheuer, 1995; Yackel, 1998; see also Consogno, Boero, & Gaz-zolo, 2006); and the importance of argumentative skills in curricula aimed at enhancing students’ intellectual autonomy (see Maher, 2002). On the other hand, what we know about argumentative skills (starting from Piaget, 1923, 1947) implies that they cannot be developed within the narrow borders of one discipline (in particular, mathematics): students’ argumen-tative potential needs to be nurtured across a lot of different activities, demanding a large amount of time. The contributions by Yackel, Maher, and others mostly concern classroom activities about mathematical subjects in grades 3 or 4 and onwards. How can teachers root these argumentative activities and initiate them in earlier stages? We think that it is neces-sary for educators, especially during early grades, to change their focus from mathematics to a wider intervention to develop argumentative skills that are relevant for mathematical argumentation.

Our hypothesis here (supported by experimental evidence) is that early development of argumentative skills can be favourable to mathematical argumentation as it fi ts into a suitable management of the appropriation of written language by fi rst grade students. This hypoth-esis offers an interesting perspective on the intervention on students’ argumentative skills. Indeed, enabling students to write texts is a crucial goal for primary school teachers. Synergy between the achievement of this goal and the development of students’ argumentative skills, relevant for mathematics education, is a possible outcome of this part of the study.

The introduction of students to written texts

This can be schematized as a three phase activity: students go from rough oral expression (or written, depending of their abilities) concerning some experience in class to production of an organised discourse as a text prepared to be written down (or rewritten), then to the fi nal written text. The children must fi rst produce their own discourse, then reconstruct verbally the coherency of the events into what we may call an “oral text,” and fi nally write down the organised text. Let us note the alternation of presence and absence of a real “other” who regu-lates the production of the written text. This “other” is usually the teacher, but it gradually becomes a role for other students, too.

The theoretical sources and their coherence

General theoretical positions (Vygotsky, 1992, chapter 6; Duval, 1999) inspired educational choices concerning the approach to writing, and further refl ections provided us with hints about the characterizations of argumentation and the hypothesis relating mathematical argu-mentation to the development of written texts. The experiments provide evidence of:



The effi ciency of our characterization of argumentation (argumentative skills appear very partially at the beginning, but develop through school years);The effects of the development of written text on students argumentative skills; andThe development of students’ skills in mathematical argumentation.

Concerning the relationships between oral and written texts

There are two main references, one coming from Vygotsky’s seminal work about the child’s transition from oral communication to written text, the other related to Duval’s investigation about the specifi city of the written text in comparison with the oral text. Briefl y, in Vygotsky’s elaboration (see Vygotsky, 1992), the transition from oral to written text is considered as a prototype for the transition from common knowledge to scientifi c knowledge in terms of consciousness, intentionality and systematic organisation.

Duval (1999) highlights some characteristics of cognitive processes underlying writing (as compared to oral communication):

writing requires more conscious control than speaking;writing requires a reorganisation of the oral text; andwriting allows one to escape the constraints of the oral text (as concerns memory, evidence, etc.).

Vygotsky’s and Duval’s contributions are relevant in order to legitimate the hypothesis that the transition from oral to written texts can be an occasion to develop argumentative skills relevant for mathematical activities.

Concerning social interaction in the classroom

We consider social interaction as it works in communication situations. Communication is important as an act of social belonging of a student both to the classroom life and to the class-shared knowledge construction. In particular, it can be seen as a way of developing linguistic representations of knowledge. These aspects have been widely considered in the literature over the last two decades (see Steinbring et al., 1998, for a representative set of orientations in the fi eld of mathematics education). We consider communication as a condition for cultural development of the individual (see the fi rst two episodes in this section) and for the group of which one is a part (see the third and fourth episodes). Indeed, communication refl ects and infl uences the development of thought (see Vygotsky’s comments about Piaget’s internal lan-guage; Vygotsky, 1992, chapter 2). In particular, argumentation in communication situations can enhance the development of students’ argumentative skills.

Concerning context

Wedege (1999) proposed a distinction between “situation context” (e.g., workplace, class-room social context, computer learning environments, etc.) and “task context” (e.g., everyday life situations evoked in a problem-solving task). We may observe that, although these two meanings of “context” (“task context” and “situation context”) are, in principle, easy to distinguish, many studies that regard one or the other actually deal with both—this happens for precise reasons concerning the studies’ theoretical orientations. Some important theoreti-cal orientations underlying studies about “situation context” in mathematics learning (e.g., the “situated cognition” perspective) concern the social construction of knowledge related to common, shared, everyday life experiences. On the other hand, many studies concerning the opportunities offered by “task contexts” are coherently conceived from a perspective of

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“social construction of knowledge” (for an example, see Bartolini Bussi et al., 1999). Here, we are interested in both “situation context” and “task context,” paying special attention to the role of suitable “task contexts” in the development of argumentative skills.

The main hypothesis

Previous theoretical considerations legitimate the following hypothesis: interactive manage-ment of students’ approaches to writing can provide students with the opportunity to develop argumentative skills, which are relevant for mathematics education, provided that the follow-ing conditions are fulfi lled:

Classroom discussions and one-on-one teacher-student interactions orchestrated by the teacher are aimed at transforming students’ utterances into pieces of written texts through explicit motivations by the teacher (and/or more competent peers) for the changes to be performed; andSuitable tasks are chosen, based on concrete operations related to familiar task contexts that ensure the possibility of immediate feedback for students’ mistakes and incomplete texts.

Experimental evidence

The following three episodes come from fi rst-grade classes that were part of the Genoa Group Project. The fourth episode comes from a third-grade class: it shows how the argumentative skills developed since the fi rst grade can intervene in a discussion concerning a space geom-etry situation.

First episode: End of January, Grade 1. Interaction between the teacher (T) and Maria (a low level student who is approaching the point when she can produce written texts).

Students have already learned to report orally some easy, concrete procedures.

T: Tell me, Maria, how you have produced the soap bubbles?Maria: I put the soap solution in the glass, then I blew within the glass, and the bubbles

came outT: But if you blow within the glass, no bubble comes out. Do it!Maria: I have forgotten to say that I blow within the soap solution through a straw.T: OK, now you can tell me with precision how you have produced the soap

bubbles.Maria: I put some soap solution in the glass, then I took the straw and I blew through it,

and the bubbles came out.T: (repeats Maria’s phrases) OK, now you can dictate your text to me. Remember that

you must speak very slowly, so that I have enough time to write.(Maria dictates a text that is very close to her oral text, and the teacher writes it down.)T: (slowly reads the whole text, then concentrates on a particular sentence): Maria, pay

attention: “I took the straw and I blew with it.” If I take the straw and I blow with it (the teacher performs the action), no bubble comes out!

Maria: I have not said that the straw was put within the glass...T: Within the glass?Maria: No, within the soap solution!T: Did you put the whole straw within the soap solution?Maria: No, only the end!T: OK, now you can dictate the right sentence: “I took the straw, I put its end within

the soap solution in the glass, then I blew and the bubbles came out.”(Maria repeats the sentence with no substantial change).

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Comments about this episode

The episode represents a fi rst level of attainment of awareness about some rules that written texts must have (completeness, precision, inner ordering, etc.). Argumentative skills are pres-ent in a very limited way on the student’s side (during the interaction with the teacher—for instance, see her second intervention). No argumentation is produced by the student. From this point of view, this episode represents only a preliminary, though important, step towards the development of argumentative skills. Let us consider the whole dialogue as a discourse. The teacher maintains a line of discourse comparable to an argumentative one: for instance, all the data (elements of the experience) that played a role had to be made explicit and stated in the right order, and logical reasons were given by the teacher to justify his demands. Some of these reasons may be seen as verbal logical reasons, some as related to the “logic” of the events. Therefore, we can say that this sequence is not only an introduction to the production of written texts, but also to some rules of argumentation.

Second episode: End of March, Grade 1. Students have to plan how to assemble a toy wind-mill. They can see a toy windmill already assembled and the pieces to be assembled to get another toy windmill. Stefania is an average-level student. She is already able to write texts, and the one-on-one interaction with the teacher concerns a text written by her.

T: Let us try to do it. I take the nail and I put its tip into the wood stick...(teacher mimics the action)Stefania: It does not work... How can I put the propeller?T: Explain why your text does not work.Stefania: Because if I put the nail into the wood stick, then I cannot put on the propeller,

because the head of the nail prevents the propeller from going over the nail. And also the washers cannot go over. I must put on one washer, then the propeller, then the other washer... Then I can put the nail into the wood stick.

T: OK, write a new text, including the explanation of the reason why you need to postpone putting the nail into the wood stick.

(Stefania’s text: I will put one washer over the nail, then the propeller, then the other washer. Then I will put the nail into the wood stick. If I had put the nail into the wood stick at the beginning, it would have been impossible to put the washers and the propeller over the nail.)


The episode shows how, in a fi rst-grade class, the teacher can provoke awareness about logi-cal requirements concerning a space-time situation. The teacher suggests an argumentative style to the student, and the student internalises it when writing the second version of her text. Here the student produces an argumentation that involves skills that are relevant from a mathematics education perspective. Reasoning and metacognitive reasoning are consciously dealt with by her; although this reasoning is prompted by the teacher, the teacher passes to her some of the responsibility to lead the argumentation: she has to explain why her text does not work. The teacher helps her fi nd out why it does not work; it is as if she suggested a counter example through material facts. The student puts it in words and draws conclusions about the structure the text should take, and, on the cognitive side, she becomes more conscious of the role of elements of the experience, comparable to the necessary data and backings of an argumentation.

Third episode: End of March, Grade 1. This is a classroom discussion about how to prepare a report for an absent schoolmate and some activities that concerned temperatures.

T: During the last week Fatima was ill; now she has recovered, and in two or three days she will be back with us. Last week we did a lot of work with the thermometer. It would be good to prepare a text to explain to her what we have done, and why.



Remember: we must explain why we decided to use the thermometer, and how. Her mother will help her to understand the text, but her mother was not here, so it is necessary to be extremely clear in preparing our text. What should we write in this text?

(As usual, in the construction of a synthesis about classroom work, the teacher goes to the big blackboard and writes down what students propose.)

Debora: Dear Fatima, we have learned to use the thermometer.Ugo: But Fatima does not know why we have decided to use the thermometer.Bianca: We could write: We have decided to use the thermometer to get a true... to be able

to say if it is true... (Here the meaning of a tool, or its contextualization intervenes. This aspect is very important in

a mathematical argumentation to relate procedures or tools to activity, argument of adequacy of a tool to an activity.)

Ugo: That today is warmer than yesterday.Daniele: Because there was somebody who said that it was warmer, and somebody who said

that it was colder.(A fi rst step in the process of modelling; we can identify the role of mathematical tools in general

reasoning.)Luca: And somebody said that it was warm, while I said that it was rather cold.T: Now we can try to write the fi rst sentence about why we have decided to use the

thermometer.Debora: We have decided to use the thermometer in order to establish if it is really cold or

warm, out of our impressions.Ugo: And if today is colder or warmer than yesterday.(The teacher writes the whole sentence containing Debora’s and Ugo’s contributions on the black-

board, she reads it very slowly.)T: But... we must compare... we must only establish that today is warmer or colder

than yesterday... And next Monday?Daniele: Today ... not only today, any day whatsoever the problem of generality of a proce-

dure, of an argument, and the expression of the generality.Patrizia: Otherwise Fatima could imagine that we have done a lot of work only for one

single day the meaning of generality.Matteo: It is.. it is for all days: Monday, Tuesday, ...T: How to modify our sentence? (she reads the sentence slowly)Ivan: (he reads the text on the blackboard): We have decided to use the thermometer in

order to establish if it is really cold or warm, out of our impressions, and if ...if one day is colder or warmer than yesterday...

Daniele: Not, not yesterday... the day... the day that comes before...Ugo: Like yesterday for before today, and the day before yesterday for yesterday.Daniele: The preceding day (systematisation of the fi nding or the procedure).Ugo: Yes, the preceding day.T: Daniele, please, say the whole sentence.


This fragment shows how the task of producing a classroom report (in a co-operative style) about the “why” and “how” of a shared activity (with an emotionally shared communication purpose) can provide students with the opportunity for an intensive argumentative activity: the elaboration of the text and the shared production of a sentence that carries generality. The fi nal text demonstrates rich argumentative skills that are relevant for mathematics education, as well as mathematical skills related to modelling.



Fourth episode (third grade class): In this situation, a mathematical argument is developed in a non-mathematical context.

The classroom discussion concerns how to set a wood table on the ground so that it is horizontal. Discussion starts by considering text written by student Daniele (copied by the teacher on the blackboard):

I put the spirit level on the table, then I look at the bubble: if it is at the center of the glass, then it means that the table is horizontal. (Daniele)

T: We must help Daniele to improve his text by explaining his mistakes and lacks, and suggesting appropriate changes. The fi nal text will be used to explain our work to another class that does not have enough time to do all the activities performed by us.

Rosa: Daniele, it is not suffi cient to check if the spirit level is horizontal... It can be hori-zontal, and the table is not horizontal.

(Rosa contradicts the statement. She is able to fi nd the right expression to oppose it, and it is logi-cally correct.)

Daniele: But we have seen that if we put the spirit level on the table, and the spirit level is horizontal, then the table is horizontal.

Federico: Let us try again... (he takes the table and puts it on the fl oor). Here is the table on the fl oor: the table is horizontal and the spirit level... Look at it, the bubble is perfectly in the middle of the glass... The spirit level is horizontal!

(An attempt to verify the original statement, which is a useful behavior in an argumentation, but he did not grasp the delicate logical relation and its negation.)

Rosa: But I have said that if the spirit level is horizontal, the table can be inclined. It is clear that if the table is horizontal, also the spirit level is horizontal!

Stefano: It seems to me that... I do not understand the difference between Rosa and Fed-erico: Federico says that the table is horizontal and the spirit level is horizontal... Also Rosa says that if the table is horizontal, the spirit level is horizontal.

(The problem of negation is clarifi ed, extracted from the confusion of the discussion. This is also a step that has to be undertaken in an argumentation: when there is complexity, the argumentation may contain partial argumentations to fi x some steps in solving the general problem.)

Rosa: But I have said that the contrary... the contrary is not true, if the spirit level is hori-zontal... the table can be inclined. Look at this (she slowly tilts the table, in order that the bubble remains at the centre of the glass).

(Giving examples and counterexamples is part of the clarifi cation of the problem and constitutes an important step in an argumentation. Here, the steps of reasoning are logical and non-trivial, and the arguments are mathematical even though the language is not specialized.)

Do you see? The bubble still is in the centre... the spirit level is horizontal, but the table is not horizontal!

Miriam: Rosa is right! The spirit level is horizontal, but the table is not horizontal!Daniele: But if I move the spirit level on the table, it is no longer horizontal! (Exploration of different cases of validity of the statement.) T: If Daniele moves the spirit level on the table, it is no longer horizontal!Robi: Now I remember! I had put the spirit level like this and like this (he indicates two

different directions with his hands).Daniele: Yes, I meant that the spirit level must be horizontal... must be horizontal in any

direction! T: Daniele, read your text and try to modify it in order to get a more precise text. To

work on the language level of the reasoning.Daniele: I put the spirit level on the table, then I look at the bubble: if it is at the centre of

the glass, then it means that the table is horizontal. I must say: I put the spirit level



on the table in different positions, each time I look at the bubble: if it is always at the centre of the glass, the table is horizontal.

(The teacher writes the second text dictated by Daniele below his fi rst text.)T: Two remarks. First, you have used the word “positions” (she underlines “positions”);

when you spoke, you had used the word “direction” (she writes “direction” under the word “positions”). Is it the same? Second, it would be good to explain why it is so important to use the spirit level in different positions—or directions?.

(Interpretation work: the use of the expressions have to be mastered by the student, and he must be able to come back to the detailed facts that correspond to the new statement. Interpretation is a very important aspect of argumentation, and of mathematical argumentation.)


Here we say how the preceding systematic practice of argumentation related to the production of written texts in a social interaction situation allows third grade students to engage almost autonomously in an important discussion involving relevant geometric (and logical) content. The teacher has students leave a trace of the collective argumentation in the written text: not only is the production of the text an occasion to develop argumentation, but also the produc-tion of a rather complex written argumentative text becomes an explicit target for classroom discussion. For further details about this kind of situations, see Douek, 1999; Douek& Scali, 2000. See also Bartolini Bussi, 1996; Bartolini Bussi et al., 1999 for other examples in dif-ferent task contexts.

Conclusions

Both theoretical reasons (Vygotsky’s and Duval’s elaborations about writing in comparison with speaking) and experimental evidence seem to support the main hypothesis stated in this section. An early, intensive approach to argumentative skills, which are relevant for mathemat-ics education, seems to be possible through interactive management of students’ approach to writing (provided that suitable task are chosen, based on concrete operations that ensure the possibility of immediate feedback for students’ mistakes and incomplete texts).

Research perspectives and problems about the role of natural language in mathemati-cal activities considered in the preceding sections raise some important questions related to teachers’ preparation and educational implications for classroom work. We will sketch some aspects of this problematique and some didactical implications.

TEACHERS’ PREPARATION

We have already remarked that the development of linguistic competencies in mathematical activities strongly depends on a teacher’s mediation. Therefore, teachers’ diffi culties must be taken into consideration. Some obstacles come from the widespread belief that natural language is not an effi cient tool in developing and communicating mathematical knowledge because of its redundancy and lack of precision. Many reasons are advocated to support this idea: the supposed prominence of mathematics as a formal system, the need for a purely syntactic treatment of mathematical relations, the diffi culty many students have in manag-ing and understanding natural language with a suffi cient level of precision, and (last but not least) the theory of Piaget (who considered communication as the main function of natural language). Many mathematics teachers (encouraged by current textbooks) are tempted to reduce the relevance of natural language in classroom work: Tasks are formulated primarily via images or stereotyped linguistic expressions; completely nonverbal answers (diagrams,



formulas, etc.) are allowed, and verbal explications are represented at the blackboard with signifi cant support from nonverbal tools (diagrams, schemas, algebraic expressions, etc.). Add to this the increasing number of foreign students in classrooms in many countries, with its obvious consequences: Mathematics (and mathematical formalisms) is universal, so we should try to teach it by reducing its verbal aspects. Add also the fact that poorly paid teachers (a common situation in many countries) may prefer to reduce the “wasted time” required to correct students’ homework and classroom problem solutions with a strong commitment to use technical, synthetic formalisms. Finally, the quantity of mathematical content that can be presented to students by using these formalisms is much greater than in the case of a verbal presentation (Boero, Dapueto, & Parenti, 1996).

All the reasons considered above make a different perspective (concerning the development of linguistic skills related to the use of natural language as a crucial issue in mathematics edu-cation) diffi cult to accept by both prospective and in-service teachers (see Morgan, 1998, for an in-depth analysis of constraints infl uencing teachers’ choices).

According to our experience, it is insuffi cient to produce good theoretical arguments against the dismissal of natural language in mathematics class: The discussion of well-chosen examples of students’ behaviors seems to be necessary. Discussion should put into evidence the crucial function of natural language in mathematical activities, according to preceding considerations. Discussion also should focus on the quality of students’ performance in rela-tion to their current mastery of natural language in mathematical activities.

Promoting verbal activities in mathematics classes

Even if prospective and inservice teachers accept the relevance of verbal language in math-ematical activities, the educational problem related to the choice and management of suitable classroom activities remains. We can consider the role of the teacher as an indirect mediator (when he or she selects and uses students’ linguistic productions), as a direct semiotic media-tor (when he or she provides students with appropriate linguistic expressions to fi t their think-ing processes), and as a cultural mediator (when he or she provides students with important “voices” as linguistic models of theoretical behaviors in mathematics).

Let us consider the following example. Students are asked to fi nd the side of the square that has twice the area of a given square. They are required to produce a verbal report about their trials to solve the problem. Sometimes this verbal report follows the steps of the activ-ity; sometimes it is written during the activity and refl ects the ongoing reasoning. Here is an example from a seventh-grade student:

Daniele: I think that I can double the side of the square, then everything will be double, and the area will be double.

(Daniele produces a partial drawing.) No, I didn’t get a double area by doubling the side of the square. The area becomes

four times larger. I must fi nd a smaller increase. I could take one-and-one-half the length of the side.

(Daniele draws with careful measurements and calculations.) No, it doesn’t work. It is bigger than I need. I should decrease once more. I could take

1.4 times larger. I can make the calculation without making the drawing. It is suf-fi cient to multiply 1.4 by the length of the side, and then multiply by itself.

(calculations: 1.4 × 2 × 1.4 × 2 = 7.841.)Double area means 8. I am near, but I haven’t obtained 8.

The teacher selects some texts produced by students and for each of them invites students to identify similarities and differences with their own solutions. In this way, the content and expression of the chosen text is scrutinized. Following is an excerpt taken from the discussion about the preceding text:



Sabrina: Daniele has come up with a good result. The side with length 1.4 times 2 (namely, 2.8) gives a surface that is very near to the area 8, or the double of the initial area.

Pietro: But Daniele has not reached the solution. The solution is to fi nd the square of double area.

Elena: And Daniele has put his numbers by chance, why one time and one half, and then 1.4, and not other numbers?

T: Daniele, try to explain why you chose those numbers.Daniele: I said to Elena that if you see that doubling the sides of the square gives a four

times bigger area, then you must decrease the side to decrease the area. I have chosen 1.5 because it is a number in the middle between 1 and 2 (which does not work).

The same activity is repeated for three texts (usually starting from the least successful of the chosen texts). The teacher tries to encourage students to use increasingly precise expres-sions. Then the teacher presents a long excerpt taken from Plato’s Menon: The well-known episode of the dialogue between Socrates and Menon’s slave about the problem of doubling the square (Garuti, Boero, & Chiappini, 1999).

Similarities and differences are found in the students’ productions; Plato’s text is discussed as a model of using dialogue to identify and overcome a mathematical mistake. The three phases of the dialogue are put into evidence (production of the mistake and identifi cation of it by counter-examples; attempts to overcome the mistake; fi nally, solution of the problem guided by the teacher). A discussion follows about a common mistake of students and the nature of that mistake. Finally, students are asked to produce an “echo” to Plato’s dialogue by writing a dialogic treatment of the chosen mistake. Here is an example of a high-level individual production by Daniele (the chosen mistake concerns the idea that by dividing an integer by another number, one always gets a number smaller than the dividend).

Socrates (SO): Tell me, my boy, what is the result of 15:3?Slave (SL): Five.SO: Is it smaller than 15?SL: What a question! That is clear!SO: And, yet, how much is it 20:5?SL: Obviously 4, Socrates.SO: Then is it smaller than 20?SL: Exactly.SO: Then, what can you say about the divisions?SL: I think that they are always smaller than the dividend.SO: Are you sure?SL: Yes, because “to divide” means “to break in equal parts.”SO: Now perform this division: 15:1.SL: Uhm,... it makes 15.SO: But 15 is equal to the dividend.SL: It is true.SO: Why is it equal?SL: Because dividing by one is how to give an amount to one person, it remains

equal.SO: So does your theory still work?SL: Not completely. Now I see that in some cases it does not work. SO: Are you still sure you are right?SL: Yes ... perhaps ... no ... perhaps there is one case in which the result is larger ... or

perhaps not ... My Zeus, I understand nothing!(5 minutes elapse).



SO: What is the result of 2:0.5?SL: These are diffi cult questions. I am no longer able to answer.SO: Take this square [drawing] and divide it into small squares!SL: This way? [The drawing is divided into 16 pieces by drawing three horizontal and

three vertical lines, all equally spaced.]SO: Yes, good. Now the unit is the small square [drawing]. How much is 0.5 compared

with 1?SL: One half.SO: Now make one half of the small square.SL: Done.SO: Do the same for all the small squares.SL: Just a moment. Done.SO: How many halves?SL: 1,2,3 ... 32, Socrates.SO: How many unit squares at the beginning?SL: Sixteen, Socrates.SO: Then you got a result greater than the starting number.SL: Uhm... Of course.SO: And how is one half written as a fraction?SL: Uhm ... perhaps 1/2.SO: Good! Are you able now to divide a number by a fraction?SL: Yes, surely!SO: Then divide 2 by 1/2. How many times is 1/2 contained in 2?SL: According to the preceding rule, I must invert the fraction and then multiply. OK,

it makes 4.SO: How can you represent this?SL: I’ll try ... Two squares ... [drawing]. One half twice for each [drawing]. It works:

4.SO: Good!SL: I understand: The division is not only “breaking into equal parts”, but also seeing

how many times a number is contained in another!SO: Make an example by yourself!SL: 1:1/4. [He performs and illustrates it]

By comparing the initial texts with the fi nal productions, we observed how different roles played by the teacher had left traces in the students’ productions (for further details, see Garuti et al., 1999), including an increased precision in linguistic expression with the use of appropriate terms, and the assimilation of a dialogic treatment of the mistake (according to the cultural model provided by Plato’s dialogue).

CONCLUSION

The analyses performed in this chapter present theoretical reasons and experimental evidence for the complex functions fulfi lled by natural language in mathematics, especially with regard to the interplay between natural language and algebraic language and the role of argumen-tation related to some theoretical aspects inherent in the systemic character of mathemati-cal knowledge (systemic links between concepts and deductive organization of mathematical theories).

Educational implications can be summarized by saying that these complex functions can-not be fulfi lled without appropriate instruction. Spontaneous classroom discussion and nego-tiations are not suffi cient to reach the level of sophistication and mastery of natural language



needed to use it in an effi cient way. The teacher must fulfi ll the complex role of mediation, including the exploitation of students’ individual productions on one side, and the use of cultural models on the other side.

NOTES

1. The term mathematical theories includes the usual theories based on systems of axioms and devel-oped according to the rules of deduction (from Euclid’s geometry to theory of groups); this expres-sion is largely common among mathematicians when they speak about their discipline. In this chapter, we refer to theories taught (or approached) in secondary schools in most countries: Euclid’s geometry, calculus, elementary theory of probability, and so forth.

2. The term theoretical aspects of mathematics refers (at a low level of schooling) to refl ective activi-ties about concepts (comparison of defi nitions, making properties explicit, etc). At higher levels of schooling, it refers to the use of specifi c systems of signs endowed with their syntactic rules (e.g., the algebraic language), the comparison between different proofs of the same theorem, and so forth.

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