12 Developing mastery of natural language

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1 12 Developing mastery of natural language Approaches to some theoretical aspects of mathematics Paolo Boero Università di Genova Nadia Douek IUFM de Nice Pier Luigi Ferrari Università del Piemonte Orientale, Alessandria INTRODUCTION Mathematical theories 1 and the theoretical aspects of mathematics 2 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 teaching of them are good solutions. Indeed, the former represents negligence of on the part of the school in passing scientific 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 mathematics 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 mathematics 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; Bartolini 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; Bartolini 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) studies focused on the learning processes and the related difficulties 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, reflective, 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 sufficient level of familiarity with the use of natural language in the proposed mathematical activities can they perform in a satisfactory way and fully profit from these activities. The reported teaching experiments also show how teachers must have a strong commitment to increasing students development of 262 RT58754_C012.indd 262 3/20/ :36:09 AM

2 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 mathematics 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 influenced 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 specific 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 specific 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 mathematics, but rather one of a wide set of artificial 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 scientific 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 scientific concepts). Vergnaud (1990) defi ned a concept as a system consisting of three components: 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, argumentation (i.e., the use of natural language in argumentative activities) is favourable to conceptualisation. 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. RT58754_C012.indd 263 3/20/ :36:43 AM

3 264 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 reasoning), through which mathematical knowledge is organized into theories. Production of conjectures 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 field. 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 historical 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 specific 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 further 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 mathematics must involve training in the ability to create this form and then overestimating formalism, whereas When a mathematician evaluates an idea, it is significance 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 educational 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, specific symbolic expressions, and logic organizations in mathematical activities; in particular, we consider the interplay between natural language and algebraic language, and the natural language side of the mastery of connectives and quantifiers in mathematics (see the subsections Natural and Symbolic Languages in Mathematics, and The Role of Natural Language in Advanced Mathematical Problem Solving). 2. As a flexible tool, the mastery of which can help students manage specific 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. RT58754_C012.indd 264 3/20/ :36:43 AM

4 Developing mastery of natural language 265 (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, producing 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 mathematics because (according to the above Vygotskian perspective about scientific concepts) theoretical work in mathematics includes, in particular, managing different systems of signs according to specific 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 reasoning (fourth function). THE LANGUAGE OF MATHEMATICS: NATURAL AND SYMBOLIC In this section, we attempt to clarify the relationships between ordinary language and the specific languages and notation systems of mathematics, with an emphasis on advanced mathematics education. We argue that mathematics learning involves management of different linguistic varieties (registers) at the same time and that some degree of metalinguistic awareness is required to control the notation systems of mathematics, rather than specific proficiency in single languages (cf. first 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 meanings that is appropriate to a particular function of language, together with the words and structures 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 definition has been carried out by Leckie-Tarry (1995). To achieve the specific goal of comparing the word component and the symbolic component of mathematical language, the idea of register seems suitable in comparison 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 purposes and are rarely used by people (including advanced researchers) who are doing or communicating mathematics. We argue that ordinary language plays a major function in all the registers that are significantly involved in doing, teaching, and learning mathematics. Ordinary language in mathematical registers Some difficulties 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- RT58754_C012.indd 265 3/20/ :36:44 AM

5 266 Paolo Boero, Nadia Douek, and Pier Luigi-Ferrari 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 characteristics 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 meaning 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 significantly 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 conditions. 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 proficiency and algebra learning 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 different 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 mathematical 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 fields of mathematics and to other scientific 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 Koedinger (2000), who outlined the symbol precedence model (SPM), which induces a significant 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 fit 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 specific purposes, among which we mention the opportunity of performing computations correctly and effectively. RT58754_C012.indd 266 3/20/ :36:44 AM

6 Developing mastery of natural language 267 Even if in the past algebraic symbolism was introduced as a contraction of ordinary language, and in some cases (such as = 8 and three plus five equals eight ) it may be treated like that, there is plenty of evidence suggesting that the relationships between ordinary 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 predicate 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 quantifier 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 reversal 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 marbles 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 expression 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 reflective 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 cannot be used to organize one s processes or to reflect 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 reflection and control, as in the following sentence: The number we have is undoubtedly divisible by three and by two because it is the product of three consecutive numbers and therefore (looking at the sequence of natural numbers) 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 reflection 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 quantifiers, especially in the approach to mathematical analysis. Defi nitions and proofs are frequently written with an extensive use of the symbols for quantifiers, up to the level of a kind of pseudo-logic formalization, as in the following example (defi nition of continuity at x o ; see Figure 12.1). RT58754_C012.indd 267 3/20/ :36:45 AM

7 268 Paolo Boero, Nadia Douek, and Pier Luigi-Ferrari! ε > 0, δ: 0 < δ ƒ( ) ƒ( 0 ) < ε Figure 12.1 Current writing of the defi nition of continuity. The use of quantifiers by students is strongly encouraged as well. In this case, the necessity 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 content field. We can consider the following examples. When approaching mathematical analysis, many students consider a decreasing function to be the opposite of an increasing function, 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 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 presentation of common life examples requires it, but because translation from one situation to the other and related reflective activity need natural language as a crucial tool. Argumentation Research about argumentation has been strongly developed over the last four decades. Different 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 specific, argumentative features of mathematical activities; we may quote the cognitive analysis of argumentation versus proof by Duval (1991) and more recent studies about the interactive constitution of argumentation (Krummheuer, 1995) and interactive argumentation in explanation and justification (Yackel, 1998). In our opin ion, if we want to consider the role of argumentation in conceptualization and proving, we need to reconsider what argumentation can be in mathematical activities, focusing not only on its syntactic aspects or its functions 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 inductions, 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: RT58754_C012.indd 268 3/20/ :36:46 AM

8 Developing mastery of natural language 269 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 specifically, 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 argumentation to be followed and understood (one line or several converging lines). Verbal organization 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 argumentative 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 specific reflections concerning some of the issues discussed in the previous section. The aim will be to 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. 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 significant 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 mediator between mental processes, specific symbolic expressions, and logic organizations in mathematical activities) that are well correlated with students performances in algebraic problem solving. We have also found evidence that superficial use (i.e., with little metalinguistic awareness; 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 specific semantics and conventions in mathematics, resulting in student failure. Example 1: The fi rst experiment was carried out in and involved 45 freshman 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 RT58754_C012.indd 269 3/20/ :36:46 AM

9 270 Paolo Boero, Nadia Douek, and Pier Luigi-Ferrari 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 classified according to the kind of language adopted in the arguments. Three levels were identified: 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 sentences, including conditional ones (11 students). Thirteen students left university during October (after 2 3 weeks of university courses); among them, eight were LO, and five were Ll. Twenty-five students passed the algebra exam before May 1995; among them, 11 were L2, and 14 were Ll. No LO student passed the algebra 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 proficiency in algebra. By comparison, the LO seems to be a sufficient condition for failure. For more details, see Ferrari (1996). The following year, a similar experiment 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 classified 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 influenced 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 significant increment of the number of students classified 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 communicate information partly expressed in another register. These results, if confi rmed, have strong implications for the teaching of both language and mathematics. Table 12.1 Number of students at levels 0 2 for rests 1 and 2 Level 0 Level 1 Level 2 Total First test (October) Second test (November) RT58754_C012.indd 270 3/20/ :36:46 AM

10 Developing mastery of natural language 271 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 following 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 misinterpret 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 unified way if we accept an analysis of the language of mathematics that takes into account the role of natural languages and avoid overvaluing the role of specific symbolism. First, we remarked that students do not use languages flexibly; 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 specific symbolic ones), but also the flexible use of them. In this perspective, ordinary language (which is far more flexible than specific 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 reflective 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 meanings. Of course, ordinary language alone cannot guarantee meaningful learning, but it can act as mediator between everyday experience and the specific 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 flexible tool that can be used to express different meanings (e.g., truth-functional semantics) and different logics (e.g., mathematical logic). Thus, a flexible mastery of ordinary language (which cannot be achieved by means of everyday-life experiences alone but should include scientific communication) should be a necessary step to mathematical proficiency. The available data suggest one main educational implication: the need for developing mastery 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 RT58754_C012.indd 271 3/20/ :36:47 AM

11 272 Paolo Boero, Nadia Douek, and Pier Luigi-Ferrari 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 flexibility of ordinary language and the specific needs of mathematical activities. Natural language, mathematical objects, early development into systems Recent literature has widely considered the issue of the constitution of mathematical knowledge through language, according to different theoretical orientations; in particular, for reasons 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 specific 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 representations 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 conscious use. This function of argumentation strongly depends on teacher s mediation and is fulfi lled when students are asked to describe and discuss efficient procedures and the conditions of their appropriate use in problem solving. The comparison between alternative procedures 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. RT58754_C012.indd 272 3/20/ :36:47 AM

12 An example Developing mastery of natural language 273 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 activities concerning fields of experience from everyday life (Boero et al., 1995). For instance, in fi rst grade, the money and class history fields of experience ground the development of numerical knowledge and initiate argumentation skills, as well as the use of specific symbolic representations. We consider data coming from direct observation, the students texts, and videos of classroom discussions. Grade II: Measuring the height of plants in a pot with a ruler We analyze a classroom sequence consisting of five 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 difficulty 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 specific 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 difficulties met by students were as follows. First, students found it difficult to focus on the problem. The teacher, T, used argumentation (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 field. 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 transferred 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 difficulties 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. RT58754_C012.indd 273 3/20/ :36:47 AM

13 274 Paolo Boero, Nadia Douek, and Pier Luigi-Ferrari 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 possibilities, 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 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 solution 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 orchestrating 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, effectively 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 problematic 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 RT58754_C012.indd 274 3/20/ :36:48 AM