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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.
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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.
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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
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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.
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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
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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
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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,
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“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
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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
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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.
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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.
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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).
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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
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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
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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.
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