European Research in Mathematics Education I.II: Group 7 232 THE MEANINGS OF MATHEMATICAL OBJECTS AS ANALYSIS UNITS FOR DIDACTIC OF MATHEMATICS Juan Diaz Godino, Carmen Batanero University of Granada, Facultad de Educación, Campus de Cartuja, 18071 Granada, Spain jgodino@goliat.ugr.es batanero@goliat.ugr.es Abstract: In this report we argue that the notion of meaning, adapted to the specific nature of mathematics communication, may serve to identify analysis units for mathematical teaching and learning processes. We present a theory of meaning for mathematical objects, based on the notion of semiotic function, where we distinguish several kinds of meanings: notational, extensional, intensional, elementary, systemic, personal and institutional. Finally, we exemplify the theoretical model by analysing some semiotic processes which take place in the study of numbers. Keywords: ontology, semiotics, mathematics education. 1. Introduction According to Vygotski (1934), the unit for analysing psychic activity - which reflects the union of thought and language - is the meaning of the word. This meaning conceived as the generalisation or concept to which that word refers - englobes the properties of the whole, for which its study is considered, in its simpler and primary form. We think that the search of analysis units for mathematical teaching and learning processes should also be focussed on the meaning of the objects involved. However, the notion of meaning should be interpreted and adapted to the nature of mathematical knowledge and to the cognitive and sociocultural processes involved in its genesis, development and communication. We agree with Rotman (1988) in that it is possible and desirable to develop a specific semiotic of mathematics, which would take into account the dialectics between mathematical sign systems, mathematical ideas and the phenomena, for the understanding of which they are built, within didactic systems. http://www.fmd.uni-osnabrueck.de/ebooks/erme/cerme1-proceedings/cerme1-proceedings.html European Research in Mathematics Education I.II: Group 7 233 To reach this aim, we consider it necessary to elaborate a notion of meaning specifically adapted to didactics, by interpreting and adapting existing semiotical and epistemological theories. This need was forecast by Brousseau (1986 p. 39), who wondered whether there is or there should be a notion of meaning, unknown in linguistics, psychology, and mathematics, though especially appropriate for didactics. In this research work, we present some elements of a semiotic model, specific to Didactic of Mathematics, starting out from the notion of semiotic function proposed by Eco (1979), and classifying mathematics entities into three types: extensional, notational and intensional ones. Based on the different nature of these types of mathematical entities and the contextual factors conditioning mathematical activity, we identify meaning categories to describe and explain the interpretation and communication processes taking place in the heart of didactic systems. The notion of meaning -conceived as the content of semiotic functions and applicable to mathematical terms and expressions, as well as to conceptual objects and problem situations - allows us to identify analysis units for mathematics teaching and learning processes. We finally describe some of these units, applying them to analyse examples of semiotic processes at the different stages of the study of whole numbers. 2. Meaning as the Content of Semiotic Functions We use the term ‘meaning’ according to the theory of semiotic functions described by Eco (1979), and consider the pragmatic context as a conditioning factor of such semiotic functions. Here, this context includes the set of factors sustaining and determining mathematical activity, and therefore, the form, appropriateness and meaning of the objects involved. According to Eco, “there is a semiotic function when an expression and a content are in correlation” (Eco 1979 p.83). Such a correlation is conventionally established, though this does not imply arbitrariness, but it is coextensive to a cultural link. There may be functives of any nature and size. The original object in the correspondence is the signifiant (plane of expression), the image object is the meaning (plane of content), that is, what it is represented, what it is meant, and what is referred to by a speaker. http://www.fmd.uni-osnabrueck.de/ebooks/erme/cerme1-proceedings/cerme1-proceedings.html European Research in Mathematics Education I.II: Group 7 234 In a previous article (Godino and Batanero 1998) we analysed the emergence of mathematical objects from the meaningful practices carried out by persons or institutions when solving specific problem fields. A meaningful practice is defined as “a manifestation (linguistic or not) carried out by somebody to solve mathematical problems, to communicate the solution to other people, to validate and generalise that solution to other contexts and problems ”, and the meaning of a mathematical object is identified as the system of practices linked to the field of problems from which the object emerges at a given time. Since semiotic functions are established by a person in a given context with a communicative or operative intention, for us these functions can be considered as meaningful practices and reciprocally, behind each meaningful practice we could identify a semiotic function or a lattice of semiotic functions. Meaningful practices, conceived as intentional actions mediated by signs, might be the basic units for analysing cognitive processes in mathematics education. 3. Notational, Extensional and Intensional Meanings We can establish semiotic functions between three primitive types of mathematics entities: • Extensional entities are the problems, phenomena, applications, tasks, i.e., the situations which induce mathematical activities. • Notational entities, that is, all types of ostensive representations used in mathematical activities (terms, expressions, symbols, graphs, tables, etc.) • Intensional entities: mathematical ideas, generalisations, abstractions (concepts, propositions, procedures, theories). In Godino and Recio (1998), we analyse to some extent the nature of these entities interpreting and adopting ideas by Freudenthal, Vergnaud and Dörfler. Notational entities play the role of ostensive and essential support that makes mathematical work possible, because generalisations and situation problems are given by notational systems, which describe their characteristic properties. Abstractions are not directly observable and problem situations are frequently used to provide abstract mathematical http://www.fmd.uni-osnabrueck.de/ebooks/erme/cerme1-proceedings/cerme1-proceedings.html European Research in Mathematics Education I.II: Group 7 235 objects with a context. Abstractions and situations are neither inseparable from the notations (ostensive objects) which embody them, nor identifiable with them, that is, we consider that mathematics cannot be simplified to the language which expresses it. The three types of primary entities considered (extensional, intensional and notational) could perform both roles of expression or content in semiotic functions. There are, therefore, nine different types of such functions, some of which may clearly be interpreted as specific cognitive processes (generalisation, symbolisation, etc.). In this paper, we classify and characterise these functions as regards to the content (meaning) involved, so that the nine types are reduced to the following three: (1) Notational meaning: Let us call a semiotic function notational when the final object (its content), is a notation, that is, an ostensive instrument. This type of function is the characteristic use of signs to name world objects and states, to indicate real things, to say that there is something and that this is built in a given manner. The following examples demonstrate this type of meaning: • When a particular collection of five things are represented by the numeral 5. • The symbol Pn (or n!) represents the product n(n-1)(n-2)...1 (2) Extensional meaning: A semiotic function is extensional when the final object is a situation problem, as in the following examples: • The simulation of phenomena (i.e., it is possible to represent a variety of probabilistic problems with urn models). (3) Intensional meaning: A semiotic function is intensional when its content is a generalisation, as in the following examples: • In expressions such as, “Let m be the mathematical expectation of a random variable ”, or “ Let f (x) be a continuous function”. The notations m, f(x), or the expressions ‘mathematical expectation’, ‘random variable ‘ and ‘continuous function’, refer to mathematics generalisations. Furthermore, all intensional and extensional functions imply an associated notational function, since abstractions as well as problem situations are textually fixed. http://www.fmd.uni-osnabrueck.de/ebooks/erme/cerme1-proceedings/cerme1-proceedings.html European Research in Mathematics Education I.II: Group 7 4. 236 Elementary and Systemic Meanings An elementary meaning is produced when a semiotic act (interpretation / understanding) relates an expression to a specific content within some specific space-temporal circumstances: It is the content that the emitter of an expression refers to, or the content that the receiver interprets. In other words, what one means, or what the other understands. Examples of this use of the word ‘meaning’ are the deictic signs, rigid designations (Eco 1990) where the content is indicated by gestures, indications or proper names. The content of the semiotic function is a precise object, which may be determined without ambiguity in the spatial-temporal circumstances fixed. The semiotic processes involved in building mathematical concepts, establishing and validating mathematics propositions, and, as a rule, in problem solving processes, yield systemic meanings. In this case, the semiotic function establishes the correspondence between a mathematical object and the system of practices which originates such an object (Godino and Batanero 1998). The structural elements of this systemic meaning would be the problem situations (extensional elements), the definitions and statements of characteristic properties (intensional elements) and the notations or mathematical registers (notational elements). These three types of primitive entities provide a classification of practices constituting mathematical abstractions. 5. Personal and Institutional Meanings The theoretical nature of systemic meanings and encyclopaedias tries to explain the complexity of semiotic acts and processes, but they are not fully describable. Practice systems differ substantially according to the institutional and personal contexts where problems are solved. These contexts determine the types of cultural instruments available and the interpretations shared, and therefore the types of practices involved. “Even when, from a general semiotic viewpoint, the encyclopaedia could be conceived as global competence, from a sociosemiotic view is interesting to determine the various degrees of possession of the encyclopaedia, or rather the http://www.fmd.uni-osnabrueck.de/ebooks/erme/cerme1-proceedings/cerme1-proceedings.html European Research in Mathematics Education I.II: Group 7 237 partial encyclopaedias (within a group, sect, class, ethnic groups, etc” (Eco 1990, p. 134). Due to these characteristics of the systemic meanings, we consider it necessary to distinguish between institutional meanings and personal meanings, depending on whether practices are socially shared, or just idiosyncratic actions or manifestations of an individual. In the second case, when the subject tries to solve certain classes of problems, he builds a personal meaning of mathematical objects. When this subject enters into a given institution (for example, the school) he/she might acquire practices very different from those admitted for some objects within the institution. A matching process between personal and institutional meanings is gradually produced. The subject has to appropriate the practice systems shared in the institution. But the institution should also adapt itself to the cognitive possibilities and interests of its potential members. The types of institutions interested by a specific class of mathematical problems might be conceived as communities of interpreters sharing some specific cultural instruments and constitute a first factor for conditioning the systemic meanings of mathematical objects. 6. Semiotic Acts and Processes in the Study of Numbers In this section, and to give examples of the theoretical concepts described, we apply the semiotic model outlined to the analysis of some semiotic acts and processes involved in the study of whole numbers. 6.1 Elementary Meanings: The First Encounter With Numbers The first encounter with numbers for most children, is produced at pre-school age, when their parents teach them the series of words ‘one’, ‘two’, ‘three’, etc. to count small collections of objects: hand fingers, balls, sweets, etc. Afterwards, they will find http://www.fmd.uni-osnabrueck.de/ebooks/erme/cerme1-proceedings/cerme1-proceedings.html European Research in Mathematics Education I.II: Group 7 238 school tasks close to those reproduced in Fig.1, which have been taken from a book for1st year primary teaching. The text is intended to make the child recognise and write the numerals ‘1’, ‘2’, ‘3’, ..., at the same time as different collections of objects represented are assigned to the corresponding numerical symbol. From the drawing of a head, a sun, a cat an arrow points at to the symbol 1. As an exercise, drawing 1 beside a flower and a moon is implicitly requested. A similar method is used for teaching the number 2, its form, writing, and use. In the tasks proposed, we can identify the three classes of objects and semiotic functions which characterise mathematical activity, according to our semiotic-anthropological model: Notations (ostensive instruments), extensions, and generalisations (or abstractions). Fig. 1: Learning the numbers 1 and 2 In fact, the concrete object drawings (head, sun, cat, flowers, eyes, etc.) are iconic representations of such objects; the meaning of the icons is the corresponding concrete object (extensional meaning); the schemes implicitly suggest answering the question, http://www.fmd.uni-osnabrueck.de/ebooks/erme/cerme1-proceedings/cerme1-proceedings.html European Research in Mathematics Education I.II: Group 7 239 “How many objects are there?”, so that the reference context is not just made up by mere concrete objects, but rather the problem situation of computing the cardinal. It is important to observe that, in the task proposed by the book, the physical objects are not in fact present. Therefore, the immediate reference context to which numerical symbols refer is a world of ostensive representation (textual). This fact implies additional semiotic complexity, and hence, interpretative effort by the child. In this series of tasks, we also identify two operative invariants (generalisations): • the same symbol,’1’, and the number-word ‘one’ are associated to various drawings of unitary collections; • the symbol, ‘2’, and the number-word ‘two’ are associated to different pairs of objects. The mental objects (or better, the logical entities), one and two, are implicitly evoked as from the first teaching levels. The tasks aim, in psychological terms, is the progressive construction of the objects, number one, two, etc. in the child’s mind. In anthropological terms it is the child’s acquisition of the habit of naming any collections by using the series of number-words, and the series of numerical-symbols (numerals). We also identify the following semiotic functions (acts and interpretation processes): I1: The drawing of concrete objects is implicitly related to the concrete objects (extensional meaning) they represent. I2: The object collections are interrelated with the numerical symbols, ‘1’, ‘2’ (notational meaning). I3: The number-words ‘one’, ‘two’, are associated (implicitly) with the numerical symbols, ‘1’, ‘2’ (notational type). I4: Each icon collection, and its associated numerical symbol, is implicitly interrelated with the corresponding mathematical concepts (one, two, three, ...) (intensional semiotic function). The interpretation of these semiotic functions requires specific codes and conventions which should be known and interpreted by the receiver of the message (the child), to successfully complete the tasks and to progressively acquire the notion of http://www.fmd.uni-osnabrueck.de/ebooks/erme/cerme1-proceedings/cerme1-proceedings.html European Research in Mathematics Education I.II: Group 7 240 number. We point out that the spaces left between the different icons play a fundamental role, since they inform which objects should be counted in each case. The writing guides for learning to draw the numerals ‘1’ and ‘2’ are also full of symbolisms, which they are difficult to decode without the teacher’s assistance. They graphically present rules such as: “Do this drawing in the way I show you and repeat it several times”. This type of rules poses the subject with a problem situation (or simply, a routine task), and involves an extensional meaning, according to our theoretical model. Our analysis shows the multiplicity of codes for whose recognition the children will require a systematical teacher’s assistance. This supports Solomon’ s thesis (1989 p. 160), that “knowing number should be reconceptualized as involving entering into the social practices of number use”, and not as an issue of individual construction of the necessary and sufficient logical structures for understanding numerical concepts. 6.2 Systemic Meanings of Numbers In the previous section we have shown examples of notational, extensional and intensional elementary meanings involved in the study of numbers at elementary school. The organised set of these elementary meanings would correspond to what we call a number systemic meaning, which would be personal or institutional depending on whether we take a particular subject (a child) or an institution (community of interpretants) as a reference. The systemic meaning of numbers within a given teaching level (school institution) is determined by curricular documents, school textbooks, and by the teachers’ own preparation of their lessons on a mathematical topic. We can observe that the meaning of numbers in curricular documents, is described in an encyclopaedic or systemic form, since it refers to a complex of situational, intensional and notational elements. Numerical competence will be achieved through http://www.fmd.uni-osnabrueck.de/ebooks/erme/cerme1-proceedings/cerme1-proceedings.html European Research in Mathematics Education I.II: Group 7 241 carrying out an organised practices system of progressive complexity throughout a prolonged period of time, which is extended beyond primary teaching. The institutional systemic meaning of numbers described in the curricular documents, will later be interpreted by the textbook authors and by the teachers themselves when designing their didactic interventions, to select and fulfil the practices they consider most appropriate to their institutional circumstances. These practices will finally be carried out by the pupils, and will determine the personal meanings that these pupils progressively build. At primary school, numbers are some “special symbols”, 1, 2, 3, ... associated to collections of objects, to count, order, and name them. Children may also carry out activities with concrete materials (rods, toothpicks, multibase blocks, abaci) which constitute more primitive numbering systems than the place-value decimal numbering system, privileged by mathematical culture due to its efficiency. It is not rare, therefore, that if we ask a child, “What are numbers?”, he/she will answer, at best: “They are symbols, 0, 1, 2, 3,. ...., invented by man to count and compare quantities. These symbols form the set of numbers”. At elementary school, numbers are neither ‘the cardinal of finite sets’, nor ‘the common property to all finite sets mutually coordinable. Few people (children, adults, even teachers) would provide such a description of numbers; however, they handle numbers, know to use them effectively for counting and ordering. For these people, numbers have a different meaning from that shared by professional mathematicians. Even for professional mathematicians the descriptions of numbers may vary substantially. In Cantorian mathematics, whole numbers “are the elements of the quotient set determined on the set of finite sets by the relationship of equivalence of coordinability between sets”. However, for Peano’s mathematics, a totally ordering set will be called whole numbers if it fulfils the following conditions: • Any successor of an element of N belongs to N. • Two different elements of N cannot have the same successor. • There is an element (0) that it is not a successor of any other element in N. http://www.fmd.uni-osnabrueck.de/ebooks/erme/cerme1-proceedings/cerme1-proceedings.html European Research in Mathematics Education I.II: Group 7 • 242 All subsets of N that contain 0 and contain the successor of each one of their elements coincide with N. Therefore, there is no single definition of whole number set, but rather various, adapted to problem situations, intentions and semiotic tools available in each particular circumstance. Each definition we may make for whole numbers emerges from a specific practices system, hence, it involves a class of problem situations and specific notational systems. In principle, each phenomenological numerical context (sequence, counting, cardinal, ordinal, measure, label, number writing, computation) can produce an idiosyncratic meaning (or sense) for numbers. Institutional contexts also share idiosyncratic practice systems and, consequently, they determine differentiated meanings. 7. Conclusions and Implications The idea guiding our work is the conviction that the notion of meaning, in spite of its extraordinary complexity, may still play an essential role to as a basis for research into the didactic of mathematics. We think that an anthropological approach to this discipline, as Chevallard (1992) proposes, complemented with specific attention to semiotic processes, may help us to overcome a certain transparency illusion about mathematics teaching and learning processes, showing us the multiplicity of codes involved and the diversity of contextual conditioning factors. The construct of systemic meaning postulates the complexity of mathematical knowledge by recognising its diachronic and evolutionary nature. This makes us aware of the relevance of semiotic-anthropological analysis of problem fields associated to each knowledge, their structure variables, and the notational systems used, since knowledge emerges from people’s actions when faced with problem situations, as mediated by the semiotic tools available. Hence, it may be useful in curricular design, development and evaluation as a macro-didactic unit of analysis, guiding the search and selection of representative samples of practices characterising mathematical competence. http://www.fmd.uni-osnabrueck.de/ebooks/erme/cerme1-proceedings/cerme1-proceedings.html European Research in Mathematics Education I.II: Group 7 243 On the other hand, and taking into account the complex nature of the meaning of mathematical objects, we should often focus attention on specific interpretative processes and on the inherent difficulties of the same when analysing student and teacher classroom performances. The construct of elementary meaning and the description of its various types permits us to focus attention on the implicit codes which condition acts and processes of understanding in mathematics education. This will be useful for identifying critical points, conditioning factors of semiotic acts and processes in mathematical activity and anticipating didactical actions. The meaning of a mathematical object has a theoretical nature and cannot be totally and unitarily described. Practices carried out to solve mathematical problems differ substantially according to institutional and personal contexts. For this reason, we introduce institutional meanings to distinguish between these different points of view and uses on the same mathematical object. These practices are interpreted in this article as semiotic functions or sequences of semiotic functions, analysing their types, and taking into account the nature of their content (extensional, intensional and notational). Personal meanings are built by the individual subject - what he/she learns, his/her personal relation to the object - and do not just depend on cognitive factors, but rather on the semiotic-anthropological complex in which this relation is developed, that is, on the element of meaning and the dialectic relationships between them as they are presented. The theoretical model outlined intends to facilitate the study of the relationship between personal and institutional meanings of mathematical objects. It also implies a strong support for conceiving mathematics and its teaching and learning as a social practice. Children’s learning difficulties, errors or failures, are explained by their different interpretations of each situation with respect to what the teacher intended, or simply by their lack of familiarity with the situation. “Not having entered into the social practices of a particular situation, subjects are lost about how to act, though they make the best of it”. (Solomon 1989, p. 162). http://www.fmd.uni-osnabrueck.de/ebooks/erme/cerme1-proceedings/cerme1-proceedings.html European Research in Mathematics Education I.II: Group 7 8. 244 References Brousseau, G. (1986): Fondements et méthodes de la didactiques des mathématiques. Recherches en Didactique des Mathématiques, Vol. 7, nº 2: 33-115. Chevallard, Y. (1992): Concepts fondamentaux de la didactique: perspectives apportées par une approche anthropologique. Recherches en Didactique des Mathématiques, Vol. 12, nº 1: 73-112. Eco, U. (1979): Tratado de semiótica general. Barcelona: Lumen. Eco, U. (1984): Semiótica y filosofía del lenguaje. Madrid: Lumen, 1990. Godino, J. D. & Recio, A. M. (1998): A semiotic model for analysing the relationships between thought, language and context in mathematics education. Research Report Proposal. 22nd PME Conference (South Africa). Godino, J. D. & Batanero, C. (1998): Clarifying the meaning of mathematical objects as a priority area of research in mathematics education. In A. Sierpinska & J. Kilpatrick (Eds.), Mathematics Education as a research domain: A search for identity. pp. 177-195. Dordrecht: Kluwer, A. P. Rotman, B. (1988): Towards a semiotics of mathematics. Semiotica, 72 -1/2: 1-35. Solomon, Y. (1989): The practice of mathematics. London: Routledge. Vygotski, L. S. (1934): Pensamiento y lenguaje. [Obras escogidas II, pp. 9-287]. Madrid: Visor, 1993. Acknowledgment This research has been supported by the DGES (MEC, Madrid), Project PB96-1411. http://www.fmd.uni-osnabrueck.de/ebooks/erme/cerme1-proceedings/cerme1-proceedings.html