HEINZ STEINBRING - EDUMATEC - Educação …€¦ · · 2009-02-17partly referring to semiotic...

HEINZ STEINBRING

WHAT MAKES A SIGN A MATHEMATICAL SIGN ? – AN

EPISTEMOLOGICAL PERSPECTIVE ON MATHEMATICAL

INTERACTION

ABSTRACT. Mathematical signs and symbols have a decisive role for coding, construct-

ing and communicating mathematical knowledge. Nevertheless these mathematical signs

do not already contain mathematical meaning and conceptual ideas themselves. The contri-

bution will present basic elements of an epistemology of mathematical knowledge and then

apply these theoretical ideas for analyzing case studies of two teaching episodes from ele-

mentary mathematics teaching. In this way different roles of mathematical signs as means

of communication (oral function), of indicating (deictic function) and of writing (symbolic

function) will be elaborated.

KEY WORDS: epistemology, construction of mathematical knowledge, analysis of math-

ematical classroom interaction, mathematical signs

1. THE PARTICULAR ROLE OF MATHEMATICAL SIGNS IN THE FRAME

OF EPISTEMOLOGICAL CONDITIONS OF MATHEMATICAL

KNOWLEDGE

In general, the importance of signs for human thinking is uncontested and

fundamental. Without signs, no human thinking and no mental generaliza-

tions would exist. ‘We have no ability to think without signs’ (Peirce, 1991,

p. 42) reads the famous dictum by Charles Peirce (cf. also Radford, 2001).

A general, essential identification of the role of the sign is the one that the

sign stands for something else; Thomas Aquinas has described this feature

with the famous definition “aliquid stat pro aliquo”.

This paper attempts to work out, from an explicitly epistemological per-

spective, the particular role of mathematical signs in certain essential points

of view. This is in not at all an attempt to build a general sign concept, but

to reach a better understanding of the particularity of mathematical signs,

partly referring to semiotic concepts and models of signs. This makes it

necessary to concentrate on the epistemological nature of mathematical

knowledge and of mathematical concepts, as mathematical signs serve

mainly for “recording” mathematical knowledge and mathematical con-

cepts. Seen in this way, the particularity of mathematical signs is first of all

related to the epistemological meaning of mathematical knowledge. Math-

ematical signs are mainly seen as “instruments” for coding and describing

Educational Studies in Mathematics (2006) 61: 133–162

DOI: 10.1007/s10649-006-5892-z C© Springer 2006

134 HEINZ STEINBRING

mathematical knowledge, for communicating mathematical knowledge as

well as for operating with mathematical knowledge and generalizing it. In

this sense, mathematical signs are also cultural tools (cf. Radford, 1998,

2001a; Vygotsky, 1987), which are used in communication with other per-

sons in order to develop mathematical knowledge.

In a first approach, the characterization of the role of mathematical signsrequires the consideration of two functions:

(1) A semiotic function: the role of the mathematical sign as “something

which stands for something else”.

(2) An epistemological function: the role of the mathematical sign in the

frame of the epistemological constitution of mathematical knowledge.

According to the semiotic function, the mathematical sign relates to “some-

thing else”, thus to an opposite which is usually called the object of refer-

ence (cf. Noth, 2000, p. 141).

Figure 1. The relation between object and sign.

The suggested relation between object and sign (Figure 1) raises differ-

ent questions: Which role do mathematical signs have regarding the objects

they refer to? Of which kind is the mediation between the mathematical

signs and the objects? The semiotic function thus expresses a referential

relationship, in which emphasis is put on the representational character of

the sign.

With the second, the epistemological function, the influence of the

epistemological characteristics of mathematical knowledge comes into the

game. ‘Epistemology is about the relationship between these types of en-

tities, objects and signs’ (Otte, 2001, p. 3). This function indicates the

possibilities with which the signs are endowed as means of knowing the

objects of knowledge.

Both functions – the semiotic and the epistemological – have to deal

with the object (of knowledge) which is in relation with the (mathematical)

sign. This perspective emphasizes the problem of the ontology of the ob-

ject for which the sign stands. Depending on the conception of the nature

of mathematical knowledge, different interpretations are connected with

the relation between mathematical signs and objects. In the frame of an

empirical foundation of mathematical knowledge the object in question

could be seen as a concretely given real object, and such an understanding

of the object would determine a certain kind of relation between (mathe-

matical) signs and objects. With a theoretical conception of mathematical

MATHEMATICAL INTERACTION 135

knowledge, the object could gain a different ontological status, for instance

as an embodiment of a structure, and thus this ontological perspective would

change the relation between mathematical sign and object. In the following

the assumption is made that the particular epistemological characteristics

of mathematical knowledge have an important influence on how the medi-

ation between “object/reference context” and “mathematical sign/symbol”

is steered.

For the intention of working out the particularity of mathematical signs

in the connection of the two dimensions mentioned above, the so-called

epistemological triangle can be used as a model (cf. Steinbring, 1989,

1997). Mathematics requires certain sign or symbol systems in order to

keep a record of and code the knowledge. The outer form of mathemati-

cal signs has developed historically and is largely conventional (cf. for the

number signs e.g. Menninger, 1979). To start with, these signs do not have

a meaning of their own, this has to be produced by the learner by means

of establishing a mediation to suitable reference contexts. This mediation

is not entirely subjective and arbitrary, but in order for the signs to become

true mathematical signs, the relation to the reference objects is at the same

time determined by epistemological conditions of mathematical knowledge

as it shows fundamentally in the basic mathematical concepts (Steinbring,

1991). The connection between the mathematical signs, the reference con-

texts and the mediation between signs and reference contexts which is

influenced by the epistemological conditions of mathematical knowledge

can be represented in the epistemological triangle (cf. Steinbring, 1989,

1991, 1998):

Figure 2. The epistemological triangle.

By means of the epistemological triangle, a semiotic mediation be-

tween “sign/symbol” and “object/reference context” is modeled which is

at the same time formed by the epistemological conditions of mathemat-

ical knowledge. Furthermore, one has to pay attention to the fact that the

epistemological constraints of the mathematical knowledge exert influence

not only on the mediation between sign and object/reference context, but

also on the construction of new, more general mathematical knowledge by

means of the same mediations. Accordingly, none of the corner points of


this triangle is given explicitly in such a way that it could become a cer-

tain point of approach to a definite determination of the triangle; the three

reference points “mathematical concept”, “mathematical sign/symbol” and

“object/reference context” form a balanced, reciprocally supported system.

At the same time, this triangular scheme should not be seen as indepen-

dent from the learner. The reciprocal actions between the “corner points”

of the triangle and the necessary structures on the side of the signs/symbols

(for example mathematical operations) and the object/reference context

(for example diagrams, functional structures etc.) have to be actively pro-

duced by the learning student, in interaction with others and observing the

epistemological influence conditions. Thus the epistemological triangle is

used for modeling the nature of the (invisible) mathematical knowledge by

means of representing relations and structures the learner constructs dur-

ing the interaction. Furthermore, one can draw up an according sequence

of epistemological triangles corresponding to a process of interaction or

learning which tries to reflect the development of the interpretations made

by the subject. The subject and the social interaction are not an analytically

equal element on the level of the three components of the epistemological

triangle, but in a certain way a meta-element that is responsible for the

construction of relations in this scheme. In the ongoing development of

the knowledge, the interpretations of the sign systems together with the

selection of reference contexts are modified or generalized accordingly by

the learner.

This relation of mathematical knowledge to the learners‘ activities opens

the perspective to the more general cultural context in which the develop-

ment of mathematical knowledge is embedded. Mathematical knowledge

cannot be revealed by a mere reading of mathematical signs, symbols, and

principles. The signs have to be interpreted, and this interpretation requires

experiences and implicit knowledge – one cannot understand these signs

without any presuppositions. Such implicit knowledge, as well as attitudes

and ways of using mathematical knowledge, are essential within a cul-

ture. Therefore, the learning and understanding of mathematics requires a

cultural environment.

Symbols, symbolic relationships and the introduction into the use and the reading

of symbols are essential aspects for the formation of every culture (cf. Wagner,

1981, 1986). Mathematics deals per se with signs, symbols, symbolic connec-

tions, abstract diagrams and relations. The use of the symbols in the culture of

mathematics teaching is constituted in a specific way, giving social and com-

municative meaning to letters, signs and diagrams during the course of ritualized

procedures of negotiation. Social interaction constitutes a specific teaching culture

based on school-mathematical symbols that are interpreted according to particular

conventions and methodical rules. In this way, a very school-dependent, empirical


epistemology of mathematical knowledge is created interactively, in which the

reading of the introduced symbols is determined strongly by conventional rules

designed to facilitate understanding, but, in this way, coming into conflict with

the theoretical, epistemological structure of mathematical knowledge. The ritual-

ized perception of mathematical symbols seems to be insufficient according to the

analysis of critical observers, because it does not advance to the genuine essence

of the symbol but remains on the level of pseudo-recognition (Steinbring, 1997,

p. 50).

Educational problems in mathematics as they become visible under an

epistemological perspective in everyday teaching practice have been an

essential source of the development of the epistemological triangle.

The practice of mathematics, especially in school, is . . . usually seduced to an

identification of sign and signified by means of the atomization, the algorithmic,

as it is expressed in formulas as a procedure of calculating, or, if one makes the

threefold distinction of concept, sign and object, which would in fact be necessary,

it leads to an identification of sign and object, neglecting an independent concept

(Otte, 1984, p. 19).

The true mathematical object, that is the mathematical concept, may not

be identified with its representations.

A mathematical object, such as a function, does not exist independently of the

totality of its possible representations, but it is not to be confused with any particular

representation, either. It is a general that . . . cannot as such be exhausted by any

number of representations (Otte, 2001, p. 33).

This criticism of the identification of sign with object or even with the

mathematical concept, is also formulated in the philosophy of mathematics:

. . . in certain branches of mathematics the symbols and diagrams have often been

confused with the very mathematical objects they are supposed to denote or rep-

resent. Thus we not only take the result of manipulating numerals or geometric

diagrams or paper Turing machines as direct evidence of properties of numbers,

geometric figures or abstract Turing machines, we also tend to confuse numer-

als with numbers, drawings with figures, and paper machines with abstract ones

(Resnik, 2000, p. 361).

This triple distinction “concept, sign/symbol and object/reference context”

is an essential characteristic of the epistemological triangle that serves to

model the semiotic mediation between reference context and sign under

the epistemological influences of mathematical knowledge. The following

two statements about the epistemological triangle are essential for this

semiotic mediation from the point of view of mathematical knowledge and

mathematic-didactic questions:

• The mathematical concept is relatively autonomous and must be distin-

guished from the object/reference context and from the sign/symbol.


• In processes of developing and learning mathematical knowledge, con-

crete objects are more and more replaced by mental objects and structures

on the side of the reference object.

1.1. The mediation between sign and reference context by meansof mathematical concepts

Elementary probability theory is a paradigmatic example for the analysis

of basic epistemological problems of mathematical knowledge. In the early

history of probability, games of chance provided simple and ideal situations

in which both a direct form of chance and a concrete structuring of regular

aspects were manifested in the physical symmetry of the chance devices

and their use (cf. Hacking, 1975; Maistrov, 1974). For the historically early

concept of probability one can think of the sign system given as a concrete

case by “fraction numerals” and of an according reference context given

by the “ideal die”.

Figure 3. The epistemological triangle: The concept of probability.

Later the reference context has changed into “statistical collectives”

(R. von Mises) and the sign system described the “limit of the relative

frequency” by a mathematical expression (as for instance hn (E) ≈ p(E)

for a rather big number n of trials). And even later, the reference con-

text contained “stochastically independent and dependent structures” and

the corresponding sign system listed mathematical statements representing

“implicitly defined axioms” (cf. Steinbring, 1980).

An important feature of this semiotic structure consists in the fact that

the reference context or the object does not represent a direct, solid, and

definite pre-given product, but that, in the course of the development of

the mathematical knowledge, the learning subject gradually changes its

meaning into a structural connection. Accordingly, mathematical meaning

in the interplay between a reference context and a sign system is produced

by means of transferring possible meanings from a relatively familiar, or

partly known, reference domain to a new, still meaningless sign system.


The concept of probability has to be distinguished from the signs used

(for example P({K }) = 1/6, which can be understood simply as a frac-

tion). Nevertheless, every mathematical concept fundamentally needs some

sorts of signs in its developmental process. The signs in probability theory

are different from the objects/reference contexts (here for example dice

or wheels of fortune, then mainly mental objects, or random structures,

later also called reference contexts), but they require these in order to be

interpreted, according to the idea that the sign stands for something else.

Additionally, the concept has to be distinguished from the object or

reference context, yet it needs this “reference context” as an “exemplary

embodiment” of a structure or a relation. Furthermore, the concept can

neither be equated with the mediation between “object sign”, i.e. only this

mediation, e.g. throwing the dice and observing the event: “E = {even

number of pips falls}” and using P(E) = 1/2 as a description with signs,

expresses the epistemological concept aspects only in an insufficient way,

since the probability must sooner or later be used for the diagnosis of this

relation, and since it cannot be entirely derived from this relation.

This fundamental point of view, that mathematical concepts as theoret-

ical concepts cannot be completely reduced to other concepts nor to other

knowledge, but eventually exist independently, becomes clearly and ex-

plicitly visible for probability for the first time in Bernoulli’s theorem. In

the frame of the early classic probability, the signs used to code probabil-

ities are fractions that indicate the proportion of favorable to unfavorable

cases. However, these “ideal” measured values must be carefully distin-

guished from the “true” probabilities of a chance experiment with random

generators. With chance experiments, the probability can be estimated in a

preliminary way with the help of the empirical law of large numbers, thus

of observed relative frequencies.

According to the given situation in the epistemological triangle, one can

regard the ideal fraction numerals as examples for the vertex “mathematical

signs/symbols” in order to determine the searched probabilities. And the

patterns of relative frequencies (as empirically measured values), which

Figure 4. The mediation between ideal and empirical probability.


van be observed in the real chance experiment, can be placed under the

vertex “object/reference context”.

How is a relation established between “object/reference context” and

“mathematical signs/symbols”? How in the example of probability a rela-

tion is established between empirical and ideal probability? It would be too

simple to claim that the (ideal) probability eventually becomes identical

to the relative frequency (after very many trials), thus having an identity

between sign and object (Otte, 1984). However, this relation is not a simple

identity, it is based on a complex structure which is essentially determined

by the epistemological conditions of the probability concept. The prelim-

inary, rather direct, relation between relative frequency and classic prob-

ability in the frame of the empirical law of large numbers, is an assertion

which must itself be mathematically analyzed and described according to

mathematical models and rules. In the early history of the theory of proba-

bility, the famous theorem of Bernoulli is the first exact formulation of this

relation (cf. Loeve, 1978).

Bernoulli Theorem:

If hn is the relative frequency of 0 in n independent trials with two

outcomes 0 and 1, which has the probability p ∈ [0, 1], then:

∀ ε > 0, ∃ η > 0, ∃ n0, ∀ n ≥ n0 : P(|hn − p| < ε) ≥ 1 − η.

Altogether we have three variable quantities: first the precision of the statement

considered, which is measured by ε, then the certainty with which the statement

holds, measured by η, and, finally, the number of trials made, which is given by

n. These three parameters are mutually dependent on one another, it is possible to

fix two and then to attempt to estimate the third (Steinbring, 1980, p. 131).

This statement in Bernoulli theorem, that there is a very great probability

that the relative frequency and probability of the stochastic experiment will

be as close to each other as desired as the number of trials increases, leads

to the following consequence: The relation between the signs (the ideal

fractions for elementary probabilities) and the reference objects (the chance

experiments with the observed relative frequencies) cannot be understood

as a fixed and safe definition of the concept, but the mediation between ideal

and empirical probabilities can only take place on the basis of a pre-existing

and thus relatively independent conceptual idea of probability.

In the attempt to mediate between the ideal signs for the elementary

probability and the empirical observations, i.e. the relative frequencies in

the chance experiment, one has to presuppose to a certain extent the con-

cept which has to be defined and clarified. This epistemological problem

is known as the circularity of mathematical concept definitions (in par-

ticular for the elementary concepts of probability, cf. Borel, 1965). This


circularity in mathematical concept constructions should not simply be seen

as a deficit. On the contrary, the impossibility of constructing a mathemat-

ical concept in all its details step by step from other knowledge elements,

and therefore being in need of the pre-existence of autonomous central con-

ceptual ideas, illustrates the autonomy of the elementary probability con-

cept. This pre-existence should not be seen as a kind of realistic existence,

but as an autonomous conceptual idea that in a dialectic manner regulates

the mediation between object/reference context and sign/symbol and, vice

versa, the unfolding mediation will produce new conceptual mathematical

knowledge. Accordingly the case of the elementary probability concept

should serve here to justify the autonomy of the “mathematical concept” in

the epistemological triangle with regard to “object/reference context” and

“mathematical signs/symbols” (cf. especially Steinbring, 1980, 1988).

1.2. Mathematical signs and reference contexts as embodimentsof structures

Children in elementary school learn early that the objects presented to them

in the mathematics class – for example apples, chips or ten blocks – are

not interesting as concrete objects, but because they embody mathematical

ideas or structures. In the social context, the children acquire the ability for a

first mediation between signs and reference objects by means of examples,

for example the mediation between the sign “3” and given concrete collec-

tions of three objects. The semiotic function that signs stand for something

else, is partly known to the children and approachable without problems.

The learning student establishes relations between signs and objects and

makes interpretations about how signs and objects are to be read and to be

interpreted. Even if “external, empirical objects” stand at the beginning,

in principle it is always about mental ideas about relations and structures,

embodied in signs and reference contexts. This change from rather empir-

ical features to structural connections can be observed in the development

of the number concept. The mathematical signs are no mere “names or

Figure 5. The semiotic mediation between numbers and objects.


abbreviations for objects (of any, also abstract kind)”, but they contain

structures, patterns and relations themselves that have to be constructed

by the learner. Thus the quantity “12.38 km” is a mathematical sign with

structure which can for instance be represented by the decimal place value

table, in which every position is in relation with the others: Tens (T), Units

(U), tenths (t), hundredths (h), thousandths (t) etc. (Steinbring, 1998).

Figure 6. Sign and reference context as mathematical structures.

Since the signs /symbols as well as the reference contexts essentially

represent structures and relations, it becomes possible, and additionally

in interactive development processes also often necessary, to achieve a

“change” in the positions between sign and reference context by means

of a subjective act of interpretation. This point of view, that the role of

“sign/symbol” and “object/reference context” in the mathematical devel-

opment process is not determined beforehand, but has to be appointed by

the subject for occurring mathematical entities, is on the one hand nothing

unusual from the semiotic point of view, since also given signs can stand for

something else. On the other hand, a new problem arises by means of the

epistemological particularities of mathematical knowledge, which focus

on relations and structures. Linked with the difficulties of seeing structures

embodied in the signs/symbols or in the reference contexts often uncer-

tainties emerge about what is given as a relative familiar reference context

that allows for explaining a new sign structure. Here, the learning students

have to decide about the roles of the (partly unknown) sign/symbol and

the (partly familiar) reference context. The development of the elementary

number concept in the classroom can again serve as an example.

Figure 7. The exchangeability between sign and reference context.


The diagram of the empty number line can once be a relatively famil-

iar reference context with which the arithmetic-symbolical structure could

be partly explained; but it is also thinkable conversely that in the learn-

ing process of arithmetical concept aspects the number line represents an

unknown, new mathematical sign which could be equipped with meaning

in some of its aspects on the basis of the familiar arithmetic-symbolical

structures and calculation rules.

Particularly this aspect of an “exchangeability of the positions of

sign/symbol and reference context” represents an important characteris-

tic of the epistemological triangle. Thus neither reference contexts nor

signs/symbols are external, fixed given circumstances, but mental ideas

which embody structures. Furthermore it is important to pay attention to

the fact that the reference contexts (objects) do not simply precede the

mathematical signs temporally or logically, but also the other way round,

the signs/symbols can, by means of their internal structure in the (historical)

development, increasingly gain independence and autonomy towards the

reference contexts and thus act upon the reference contexts as well as project

a structural interpretation perspective on reference contexts. Signs/symbols

and reference contexts are then temporally equal or have equal rights, none

precedes the other. The semiotic analysis of the historical development

of the number zero is a paradigmatic example in this regard (cf. Rotman,

1987).

This attempt to model the semiotic function of mathematical signs with

the help of the epistemological triangle under the simultaneous epistemo-

logical influences of theoretical mathematical knowledge has the conse-

quence of regarding the defining characteristic for the sign – something

that stands for something else – in a new way. Mathematical concepts

are not empirical things, but they represent relations. Whatever an ob-

ject/reference context for a mathematical sign might be, it must be under-

stood as an embodiment of a relational structure and not as a kind of real

thing with perceivable properties. Raymond Duval emphasizes this specific

understanding that mathematical features cannot be directly observed.

. . . there is an important gap between mathematical knowledge and knowledge in

other sciences such as astronomy, physics, biology, or botany. We do not have any

perceptive or instrumental access to mathematical objects, even the most elemen-

tary, . . . . We cannot see them, study them through a microscope or take a picture

of them. The only way of gaining access to them is using signs, words or symbols,

expressions or drawings. But, at the same time, mathematical objects must not

be confused with the used semiotic representations. This conflicting requirement

makes the specific core of mathematical knowledge. And it begins early with num-

bers which do not have to be identified with digits and the used numeral system

(binary, decimal) (Duval, 2000, p. 61).


With this description it is not intended in this study to assume an ideal

realistic existence of mathematical objects, but the invisible mathematical

object of knowledge is interactively constructed in social environments.

The impossibility of a direct sensual perception of mathematical knowl-

edge, as it is considered as well in the epistemological triangle, and the

function of mathematical signs to relate to structural reference contexts,

where this mediation is guided by epistemological conditions, requires a

further going interpretation of the “aliquid stat pro aliquo” in the frame of

mathematical knowledge. “Mathematical signs/symbols” as well as “ob-

jects/reference contexts” are embodiments of not directly visible struc-

tures, they thus do not stand for something concrete, visible objects or

features. Hence, in the epistemological triangle, “sign/symbol” as well as

“object/reference context” stand for something else, something not directly

perceivable.

The focus in this theoretical part was mainly on the internal conditions of

the relations between signs/symbols, objects/reference contexts and mathe-

matical concepts (or conceptual ideas), as illustrated in the epistemological

triangle. The epistemological triangle is a model for making the invisible

mathematical knowledge accessible with regard to its structural character,

for describing its particularities and also for analyzing interactive processes

of constructing mathematical knowledge – thus invisible relations that are

embodied in exemplary contexts and activities. This theoretical perspective

on the internal conditions must be seen as embedded in the larger cultural

context of the interactive production of mathematical knowledge as well as

in the environments of teaching and learning mathematics. It is essentially

through mathematical activities that the dialectic relations (expressed as

arrows) between the three vertices of the epistemological triangle are con-

structed in a certain way in the course of social knowledge development.

The following more practical part will show examples from elementary

mathematics teaching and try to take into account in a more substantial

manner the specific mathematical activities within the mathematics class-

room culture that are performed to construct different sorts of relations

between the three vertices of the epistemological triangle.

2. THE INTERACTIVE CONSTRUCTION OF MATHEMATICAL SIGNS –

AN EPISTEMOLOGICAL ANALYSIS OF EXEMPLARY EPISODES FROM

MATHEMATICS TEACHING

The above considerations made it clear that the mediation between signs

and reference contexts is influenced by particular factors when it is about

mathematical signs trying to record mathematical knowledge. The kind of


mediation between mathematical signs and reference objects is dependent

on the activities of the learner (cf. Radford, 2003a) and on the epistemo-

logical conditions of theoretical mathematical knowledge. This part will

focus on the social context of the interactive construction of mathematical

knowledge. For this purpose examples of classroom episodes about man-

ifold forms of constructing signs for recording aspects of mathematical

concepts will be analyzed in the following.

In classroom communications, it is very frequent to have recourse to

the natural language-as in everyday life-as well as to possibilities of direct

showing as essential means. Also in class, these means of communication

mainly serve for exchanging and recording mathematical ideas. Descrip-

tions in ordinary language, designations by means of striking names, direct

showing and referring to something, all these are preliminary forms of

interactively produced signs to code and develop aspects of mathemati-

cal concepts. In addition, technical terms, written notations, mathematical

coding signs, variables, special mathematical symbols, diagrams etc. are

the rather professional forms of mathematical signs.

The range of forms in which mathematical signs become visible in so-

cial interaction reaches from ordinary ways of speaking and direct showing

to the standard forms established for a long time in the academic discipline.

In classroom interactions, the learners are to become familiar with different

forms of mathematical signs in the interaction, to acquire their use by means

of social participation, and not to use finished given signs according to strict

rules. For the learners the signs and the forms of their interpretation develop

by means of mediations to reference objects. Further, a generalizing use of

the signs in a certain way develops only gradually in the temporal, interac-

tive development; one cannot give the learner finished mathematical signs

in their essential meaning at the beginning of his or her process of learning.

In everyday mathematics teaching, manifold forms of mathematical

signs can be observed:

• verbal formulations, own words with exemplary descriptions

• communication by means of showing and referring (deictic)

• mathematical symbols: numerals, operation signs, letters, variables, . . .

• arithmetical exercises, equations, systems of equations, . . .

• tables, geometric diagrams, graphs of functions, . . .

These forms of different signs reach from interactive signs produced rather

spontaneously in situated, exemplary learning environments to signs for-

mats, which are conventionalized and usual in mathematics teaching. The

following analyses of classroom episodes are to contribute to a better under-

standing of the kind of the interactive production and the use of these sign

forms in the mathematics classroom especially from an epistemological


perspective on mathematical knowledge. Again, this is about observing in

which way the mediations between the different signs and corresponding

reference objects are interactively produced, how the influencing episte-

mological factors are perceived, and in which way these epistemological

conditions are considered in the interactive mediations between sign and

reference object, and how in this way mathematical knowledge relations

are interactively produced.

2.1. The construction of mathematical signs by means of direct andcomparative showing – Matthi constructs mathematical relationsin a deictic manner

The following short episode is taken from a 4th grade mathematics lesson;

the students work on number walls of four levels. The problem is to find

out in common interaction in which way the increase of a base stone by 10

changes the value of the top stone; furthermore the students are expected

to develop a justification for the operative change.

Explanation: When describing the position of the numbers on the numberwall, the stones are numbered consecutively in the transcript from bottomto top and from left to right.

Figure 8. The stones of the number wall.

Before the beginning of the episode regarded here, the following three

number walls with calculated numbers and also with magnetic chips can

be found on the blackboard (Figure 9).

First of all the number wall above with the four base stones 50, 80, 20

and 30 was completely calculated. Then the second base stone (stone # 2) in

the bottom left number wall was raised by 10, and this was additionally in-

dicated by a magnetic chip. The number wall was calculated and compared

to the number wall above. It was worked out that the top stone increases by

30. All the stones in which a change occurred are marked with a magnetic

chip. In order to work on the question of how the top stone increases when

the left base stone (stone # 1) is raised by 10 (in the bottom left number

wall), the teacher has first changed the value of this stone to 60 and put a

chip there; then a student has calculated the second row of the wall.


Figure 9. Operative changes in four-storied number walls.

Now the student Matthi is called to the board; he wants to calculate the

numbers of the third level but the teacher first asks for a justification.

199 Ma Ohm, here [points at the fifth stone (1)] is also

ten more- [points at the chip of the first stone (2)]

Figure 10.

. . .

201 Ma [C gives Ma a chip] -because here is also ten

more. [puts one chip into the fifth stone of theright lower number wall (3)] Because here as

well, because it’s ten more here, [points at thechip of the first stone (4)], here is ten more [pointsat the fifth stone (5)] than there ten more. [pointsat the upper number wall (6)] . . .

Here is the same then [points at the sixthstone “100” (1) and then repeatedly al-ternately at the first and second stone ofthe right lower number wall (2 & 3)] ,

because one cannot this here plus that, if one

that [points at the fifth stone (4)].

Figure 11.


One does not get along with this here [points alter-nately at the first and at the third stone (1 & 2)] then

there [points at the sixth stone and then at the firststone (3 & 4)] , or so. That one that. [points at the sixthstone (5)] . . . . . .

Figure 12.

And here [points at the seventh stone (1)] it’s also the

same, because that one is at the margin. [points at thefirst stone (2)]

Figure 13.

It can be observed that Matthi tries throughout the entire episode to pro-

duce mathematical signs by means of several deictic procedures that are

accompanied by short verbal utterances. In this way Matthi shows (at least)

five different ways of producing mathematical signs: (a) linguistic deictics

(“here”, “there”, etc.); (b) pointing deictics (which are gestures that ostensi-

bly show the referent); (c) “touching deictics” (different from the previous

one because the referent is materially touched with the finger); (d) the

chips (which function as an index; see Radford, 2003b) and (e) the usual

number-signs of arithmetic.

Matthi first explains why the 5th stone is raised by 10. He constructs a

relation between the 1st stone and the 5th stone by means of pointing at the

chip in the 1st stone and interpreting it as “10 more”; then he puts a chip

into the 5th stone and says once more that therefore there are “10 more”

in the 5th stone as in the 5th stone of the original wall at which he also

points above. The sign forms used spontaneously by him are similar to the

descriptions already used before; he does not name the concrete occurring

numbers and he chooses the relation “10 more”. Furthermore he directly

points at the chip with the meaning “10 more”.

This first step of Matthi‘s knowledge construction is a good example

to show in a diagram the complex interaction of the five different kinds of

sign production used here (Figure 14).

The cross table indicates for each construction step the kinds of sign

production that are used. Seemingly the contiguity of the cells of the number

walls are a “natural” basis for Matthi for spontaneously using his different

kinds of deictics and direct linguistic descriptions.


Figure 14. Matthi‘s different kinds of sign production.

Then Matthi turns towards the 6th stone. By repeated direct pointing at

the first and second stone, Matthi constructs a deictic sign again with which

he wants to describe that the 6th stone cannot increase. With very open hints

at the first and second stone as well as with the statement ’. . . because one

cannot here somehow plus that, if one that’, he indicates that the 6th stone

cannot result as a sum of the first stone (raised by 10) and the second stone.

With a further deictic sign, Matthi then refers to the addition of the first

(raised) stone and the third stone, which does not lead to the 6th stone either

(this one can only be constructed out of the addition of the 2nd and the 3rd

stone).

Finally, Matthi also wants to exclude an increase of the 7th stone with

a third deictic sign: ‘The 7th stone remains the same, because the first

(raised) stone is at the outside’. With this, Matthi intends that the raised

stone cannot be used for the addition leading to the 7th stone, thus that this

one cannot be raised.

In Matthi’s knowledge construction it is not so very interesting that he -

as expected by the teacher - illustrates the change of the 5th stone, but that he

additionally explains why the 6th and 7th stone cannot be raised. He seems

to argue “indirectly”: If the 6th stone is to be raised, the first stone with “10

more” must participate in the additive construction of the 6th stone; but the

first stone may not be added with the second nor with the third stone in

order to obtain the 6th stone. Matthi justifies the impossibility of an increase

of the 7th stone according to the “indirect” pattern, but compressed to the

statement: Because the first stone is on the outside, the 7th stone remains


the same. The understanding of this justification presupposes the first, more

detailed interpretation for the 6th stone.

The new justification pattern for the impossibility of an increase of the

6th and the 7th stone can be described in the epistemological triangle by

means of the following construction of knowledge relations. Matthi spon-

taneously employs the partly filled number wall with the found numbers,

the increase “plus 10” as well as the positions as a reference context. The

mathematical signs constructed by Matthi are mainly of deictic nature and

reflect the mathematical rules for constructing number walls.

Figure 15. Matthi constructs verbal and deictic signs.

Matthi constructs new knowledge by means of tangentially showing the

impossibility of an increase of the 6th and 7th stone through a “contradic-

tion”. An increase (change) in the second row can only take place if the

increased stone is also used for the additive construction of the second row

of the wall. On the other hand, the first stone does not participate either in

the calculation of the 6th nor of the 7th stone. Matthi’s knowledge construc-

tion is relatively open; he directly points at the special fields, but intends

the additive relations lying between them as well as the relational value

“10 more”. He emphasizes impossible additive relations and thus justifies

the impossibility to change the 6th and 7th stone.

The signs used by Matthi are of the form of own verbal descriptions

and of direct pointing at different fields. Despite of this simple and partly

preliminary form his signs, he develops “true“ mathematical signs in his

construction process in the sense that through his showing and speaking he

always intends a relation between two or more stones in the number wall.


Thus it is aimed at the internal structure of the signs developed here: With

these signs, no empirical, directly perceivable properties of the reference

object “number wall“ are aimed at by Matthi, but mathematical relations

are emphasized. Furthermore, the fundamental mathematical construction

principles for number walls – which represent the conceptual aspects here

– are presupposed in the so constructed mediation between signs and refer-

ence context and they are also actively extended to new knowledge relations

and showing that some relations in the number walls are not possible. In

this way Matthi makes use of the central epistemological condition in the

concrete environment of number walls, i.e. using a mathematical concep-

tual relation for producing a specific mediation between sign/symbol and

object/reference context and at the same time further developing this con-

ceptual relation.

In spite of their idiosyncratic character and their strong situation-

dependence, the signs constructed by Matthi are mathematical signs

for coding, representing, generalizing and communicating mathematical

knowledge.

2.2. The construction of mathematical signs as symbolizationof positions in arithmetical diagrams – Kim constructs amathematical relation with a missing number

This short sequence of lessons has been conducted in a mixed 3rd and

4th grade class and deals with the topic of “crossing out number squares”.

Crossing out number squares develop out of the addition of certain border

numbers (in form of a table; cf. Figure 16). Crossing out number squares

have the following characteristics: In a (3 by 3) crossing out number square,

one is allowed to chose (circle) any three numbers, so that there is one

circled number in every row and every column. The sum of three num-

bers determined in this way is constant-independent of their choice (cf.

Figure 17). This number was called the “magic number”.

Figure 16.Figure 17.

Figure 18.

In the previous lessons the children have observed the constancy of the

sum with given squares with the help of the so-called crossing algorithm;

then procedures for producing crossing out number squares out of addition


tables have been discussed and the connection between the crossing sum

and the margin numbers has been examined.

In this lesson the children have to work on the following problem: How

can one fill a gap in a crossing out number square with a missing number

in such a way that the crossing out number square is re-established? (cf.

Figure 18). Different strategies are developed: The number 15 is mentioned

because of its visible arithmetical pattern; then possible margin numbers

are reconstructed (cf. Figure 16). During the course of this lesson Kim

makes another proposal; she wants to use the magic number (the constant

sum of three circled numbers) for determining the missing number. With

her own words – that other children have difficulty understanding – she

formulates her proposal.

Later in the lesson, Kim is asked to concretize her proposal. First she

uses the procedure to calculate the magic number by circling and crossing

numbers in the square. She circles the following possible numbers and

calculates the magic number with the task “13 + 15 + 17” .

147 K And then one could already do it this way. One

circles the fifteen [points at the fifteen in thefirst line] and this fifteen [points at the fifteen inthe second line] and adds them. And then one

still calculates how much there must be up to

forty-five.

Figure 19.

Then Kim repeats in her own words her proposal for calculating the missing

number.

With her own words, she constructs a verbal sign relating to the reference

context “number square with gap and with calculated magic number”. With

this sign, she expresses a relation between the sum of the two 15s (on the

diagonal) and the magic number 45 in a yet unclear way. The teacher then

asks Kim to concretize her suggestion; she has wiped out the markings

of circling and crossing out so that the procedure to determine the magic

number can be carried out again. Kim gives the following explanation:

161 K First one calculates, one first calculates these

numbers, that I have, which are there, what

is their result. And then. . ., and then one

calculates. . .

Figure 20.


162 L So. Now Kim explains how it goes on! Kim.

Figure 21.

163 Yes. One first calculates the numbers there, that

are there. [Kim points with her open left handglobally at all numbers in the square]

Figure 22.

165 K These three, oh, yes, this, this and then after-

wards one calculates fifteen [circles the fifteenin the second line], one takes this way. Cross

out that, and that. And cross out that, and that.

[crosses out numbers which are in the same lineor column as the fifteen]

Figure 23.Then one takes the fifteen. [circles the fifteenin the first line] Crosses the seventeen and the

thirteen. [[crosses out the not yet crossed num-bers which are in the same line or column asthe fifteen]

And then one circles this here, this here. [circlesthe empty field] And then one has to calculate

fifteen and fifteen. This makes thirty; how much

is left up to forty-five?

On the teacher’s inquiry, Kim writes down the following supplementary

exercise below the addition exercise:

166 T Just write that down like this as an addition ex-

ercise! As you just did it, with a gap if you

want to. [Kim wants to wipe out the exercise“13 + 15 + 17 = 45”] No, just leave that be-

low!

Figure 24.

At the beginning (line 161) Kim repeats the necessity of the calculation

of the magic number in a verbal, partly unclear way: ’. . . First one calculates,


one first calculates these numbers, that I have, which are there, what is

their result’. Then she begins concretely with application of the crossing

out algorithm to the numbers 15 (in the second row) and 15 (in the first

row). By means of this procedure, a visible system of mathematical signs is

gradually produced in the reference context through the graphical signs of

circling and crossing out which corresponds to the familiar, visual scheme

of the algorithm for the magic number.

With the statement here.’, ’And then one circles this here, the crossing

out algorithm is intentionally applied to an (unknown) third number, thus

the missing number on the empty field. With the second statement ’. . . then

one must calculate here, fifteen and fifteen are equal to thirty, how much

is still missing up to forty-five.’, the calculation of the magic number out

of the three circled numbers is intended on the one hand: 15 and 15 are 30.

But one cannot continue calculating with the third circled position. On the

other hand, it is now to be calculated “reversely” with the unknown position

in such a way that the number how much is missing from 30 up to 45 (the

magic number) comes into this position. The “scheme” of the calculation

of the magic number is transferred to the empty position and thus a new

sign is intended and also represented iconically. With this “sign”, the magic

number cannot be calculated in the usual way; in addition the magic number

is already known. Thus the calculation in the scheme is interpreted in a new

way: The number, which is still missing up to the magic number, comes

into the empty position.

With the teacher’s assistance, Kim writes down the addition exercise

in order to determine the missing number: “15 + 15 + = 45”. Thus it

is intentionally referred to the crossing out algorithm for three numbers

and at the same time to the “unknown” number in the empty field; in this

way a new (open) sign is constructed and written down with mathematical

notations.

From an epistemological point of view one can notice that Kim con-

structs essentially new knowledge in her argumentation with the operative

treatment of the unknown number in the empty field. One can not only cal-

culate with concrete numbers, but also transfer the scheme of the crossing

out algorithm to arbitrary fields-with numbers, but also without numbers.

Then one can operatively vary the calculation if-except for the unknown

number-all further numbers in the calculation are known. Kim already rep-

resents her argumentation in a certain “logical” sequence. First, the magic

number is determined from three possible crossing out numbers (the pur-

pose is not yet predictable); then the algorithm is applied to two numbers

and the empty field, and in order to do this, one eventually requires knowl-

edge of the magic number (here the necessary calculation of the magic

number becomes reasonable).


Figure 25. Kim constructs graphical and written signs.

In summary, the new constructed mathematical knowledge in Kim’s

argumentation can be characterized with the epistemological triangle

(Figure 12). The new relation (the unknown number or “variable”) is sym-

bolized in two ways, once in the crossing out number square as “circled

number” to determine the magic number, then in the calculation as “miss-

ing term of a sum” in an addition exercise in which the magic number to be

determined-the result-is already known. In this representation context one

works with reference to the situation with a “mathematical unknown”; this

unknown number (the missing number) is constructed as new knowledge

in a mathematical relation (as the new mathematical object) and tied into

the structural relations of the crossing out number square.

Kim uses the known graphical signs of circling and crossing out; she

transfers this graphical code to a more general situation so that now also an

empty field is circled. Thus the status of circling concrete numbers changes

to positions in the number square which stand in a relation to each other.

This generalizing use of the “circles” leads to “true” mathematical signs

since it is the relation between the circled positions that is emphasized.

Once again, the internal structure of the used graphical signs becomes vis-

ible. These signs not only mark concrete numbers to calculate with, but

more generally the possible positions of numbers (which one could under-

stand as preliminary variables) and thus mathematical relations. Besides the

graphical sign construction, a written sign construction with the equation

“15 + 15 + = 45” is produced. Compared to the usual calculation exer-

cise “15+15+15 =?”, this equation also represents a generalization from


calculating arithmetics to algebra modeling relations (cf. Winter, 1982).

The written sign system “15 + 15 + = 45” represents also an internal

structure in the form of a “pre-algebraic” equation, and not only an abbre-

viated procedure which represents procedural steps in order to calculate

numbers out of given numbers.

In thus constructed mediation between the graphical and written signs

and the reference context of the given number square with gap, funda-

mental mathematical relations in crossing out number squares-which rep-

resent the conceptual aspects here-are applied in a presupposing manner

and also actively extended to new knowledge constructions. The presup-

posed knowledge of important aspects in the fundamental relation which

is contained in the algorithm to determine the magic number, together with

a generalizing view on arithmetical calculation exercises are conditions

for Kim to be able to produce a mediation between the constructed signs

and the reference context, which has made possible the construction of

true mathematical signs. In her knowledge construction in the environment

of “crossing out number squares“ Kim uses the central epistemological

condition, i.e. taking a mathematical conceptual relation for producing a

specific mediation between sign/symbol and object/reference context and

at the same time further developing this conceptual relation.

3. CLOSING REMARKS: INTERACTIVE AND EPISTEMOLOGICAL

ASPECTS OF THE PRODUCTION OF MATHEMATICAL SIGNS

The conditions, which influence the process of mediation between sign and

reference context by means of the epistemological particularities of mathe-

matical knowledge, stand in the centre of the question of what makes a sign

a mathematical sign. The attribute “mathematical sign” or “mathematicalsymbol” is not reduced to the mere enumeration of the particular forms of

notation, ways of writing and symbol chains (as one can find it for example

in Davis/Hersh, 1981, p. 122 ff.). Mathematical signs are first of all also

very general signs, which are indispensable means for human communi-

cation and for human thinking. Just like signs in general, mathematical

signs also are carriers of knowledge and elements of communication with

others. Mathematical signs also appear in different forms in communica-

tion connections. In the examples of classroom scenes, the following signs

forms became visible, for example: speaking, showing, drawing, writing

down, noting number sentences etc. In addition to the “evolving interpreting

games” (Saenz-Ludlow, 2001, p. 20 ff) in which the students in mathematics

instruction deal with and interpret mathematical signs, the epistemological

dimension of mathematical knowledge furthermore requires a perception of


the epistemological conditions of the knowledge as relatively independent

factors opposed to the communicating subjects; the epistemological con-

ditions essentially contribute to the mediation between sign and reference

context.

. . . in epistemology as well as in logic or communication theory we have to ac-

knowledge that communication and knowledge are possible only when there is

some Other, which does neither fuse with the Subject nor is totally different from

it (Otte, 2001, p. 5).

In the previous sections, the aim was to characterize the particularity

of the sign as a mathematical sign by means of two essential aspects. (1)

In the frame of mathematical knowledge, the sign as well as the reference

object-for which the general sign stands-are always already embodiments

of something else, of structures and relations; (2) the mediation between

sign and reference object requires the consideration of a presupposed, rela-

tively autonomous existence of mathematical concept relations which take

part in this mediation process. In active, mathematical learning processes,

such mediations between sign and reference object are carried out by the

participating subjects in direct interaction; and the question is raised how

it is related to the epistemological conditions of mathematical knowledge.

The generality of the socially produced mathematical signs depends among

other factors on the question whether it is about direct, face to face commu-

nication, or whether a mathematical communication is led with absent part-

ners (cf. Radford, 2003b). In temporally longer, socio-historical processes

of developing mathematical knowledge, more and more conventionalized,

fixed and obligatory forms of mathematical signs will emerge by means of

mediations between signs and reference objects.

An important problem consists in understanding how signs are produced

already in elementary, initial mathematical learning processes, and whether

this is at all possible already in the beginning. A very essential statement is

that neither the reference context nor the signs themselves can be equated

with mathematical knowledge, that is with conceptual relations. This is

on the one hand a difficulty of “grasping“ mathematical knowledge. On

the other hand, this problem contains the possibility of grasping concep-

tual knowledge in relative independence from the form of the signs. One

must see that signs are unavoidable and indispensable for recording math-

ematical knowledge, however the kind and the form of the signs are not

unequivocally fixed. One does not necessarily need the typical algebraic

signs such as letters and operation symbols, to record algebraic relations,

as these signs themselves are not the algebraic knowledge in question. To

a certain extent one is free to choose other sign forms, and these possi-

bilities of choice serve for the change, development and optimization of


mathematical signs a suitable characterization of the invisible conceptual

mathematical knowledge.

The interaction processes which could be observed in the episodes take

place in situated learning environments with concrete features and arith-

metical connections. Children must and will construct their mathematical

signs in this situated context with their own designations and with refer-

ence to the situated features. They do not have at their disposal any finished,

general and suitable signs-for example letters-with which the general struc-

tures in the arithmetical problem fields could be constructed and justified

without problems. The fundamental statement that conceptual knowledge

and the signs for its coding must be distinguished also negates such a

possibility.

The signs of speaking, showing, drawing and writing down arithmetical

sign chains employed by Matthi and Kim are thus generally, current means

of communication in mathematics instruction. In order to become mathe-

matical signs, it must become clear in the children’s sign construction that

these signs refer to structures (of invisible mathematical knowledge), and

that conceptual knowledge relations are presupposed and applied.

Matthi’s signs are essentially of a deictic nature; in addition he speaks

supportingly of “here”, “there” and “this” when pointing at positions in the

number wall (compare with the distinction between “deictic function” and

“generative action function” made by Radford (Radford, 2000, p. 247)).

The showing of positions contains possibilities of generalization, since

the concreteness of single examples of numbers is not emphasized and

since one could imagine that the connections, which Matthi shows be-

tween different positions, could also be valid for other numbers. Important

aspects making Matthi’s deictic signs mathematical signs are the relations

between the positions in the number wall intended by his showing and the

comparison with the calculated original wall (above). Showing is in the

first approach something direct, and Matthi points with his finger or his

hand at a concrete object, a stone in the wall or a whole wall. Here the

direct showing changes to comparing showing with which relations are

intended; the showing movement back and forth between two stones of the

wall, supported by statements such as ’. . . that here not somehow plus that

. . . ’ becomes a sign for a mathematical relation. The production of such

relations between stones of the number wall by means of showing move-

ments is supported by the construction relations for number walls; only

such comparisons, which are compatible with these construction relations,

make sense.

In the learning environment “number walls”, Matthi’s production of

deictic signs which express not a direct, but a comparing showing was

probably supported by the fact that the problem is founded on an operative


connection between change of the base stones (“10 more”) and the

following consequences for the goal stone. Thus the relation “10 more”

itself must be paid attention to besides the numbers in the wall. Thus

Matthi’s deictic signs become beginning mathematical signs, which relate

to invisible structures in the mediation with reference contexts, where this

mediation is essentially contributed to by the construction conditions of

number walls.

Matthi‘s sign construction carried out in the direct interaction with the

teacher can be identified as a contextual generalization with the concept

worked out by Radford (Radford 2003b); but also under the contextual

dependence this sign construction is about the beginning of a true math-

ematical sign, where the observable change from a direct to a comparing

showing is an important indicator.

Kim‘s sign construction begins with a verbal description and a global

pointing at the number square with a missing number. With this description

(lines 161 & 163) she firstly repeats that the magic number can and is to

be calculated. Concretely she then gradually produces a graphical sign as

it is already partly familiar to the children: three circles with numbers in

them and additionally 6 crossed out numbers. This graphical sign, how-

ever, has been transferred to a new situation, as the initial requirement that

one number in the square must be circled is generalized to circling one

field in the square in which there could stand a number or not. In this way,

Kim again constructs a new, graphical sign out of the familiar sign, which

is generalized by means of this procedure of transferring. This construc-

tion of the sign also includes verbal statements ‘And then one circles this

here, here. [circles the empty field]’. Kim explains verbally which func-

tion the circling of the empty field has: ‘And then one has to calculate

fifteen and fifteen. This makes thirty; how much is left up to forty-five?’

(line 165).

The newly interpreted graphical sign which emerged through circles and

lines in the 3x3 square and is supported by Kim’s verbal remarks, is then

to be written in an already conventionalized form “as an addition exercise”

according to the teacher’s request. Here shows again a changed interpre-

tation of partly known sign representations of arithmetical exercises. The

“common” calculating exercise structure “13 + 15 + 17 = ?” where the

input numbers stand on the left side of the sign of equality and the output

numbers on the right is now interpreted in a new way: “15+15+ = 45”,

thus into a true equation which expresses a mathematical, structural relation

(cf. Winter, 1982).

The graphical sign and the written arithmetical equation, which Kim

produces in the interaction with the teacher, are true mathematical signs in

the core, as they reflect structures in the arithmetical problem field “crossing


out number squares”. Additionally, the creation of a productive mediation

between these signs and the reference context of the given square filled with

8 numbers requires beforehand part knowledge and meaningful application

of arithmetical relations which regulate the construction of crossing out

number squares. Thus the two essential features for mathematical signs

are given in this construction process.

Again a sign construction is carried out in the interaction between Kim

and her teacher, which is strongly dependent on the contextual conditions

of the arithmetical situation. On the one hand the graphical sign is first

bound to the procedure of determining the magic number, for which it was

developed and so far only used. Then it more and more becomes a situ-

ated, graphical relational designation of possible structures in the crossing

out number square in which relations between circled numbers and magic

number are transferred from concrete numbers to general fields in the ge-

ometric square and thus signal an independence from the chosen concrete

numbers. The production of a general arithmetical equation then represents

a step of getting loose from the contextual aspects to a beginning, relatively

autonomous symbolical designation of the mathematical sign worked on

here. With the conception worked out by Radford one can firstly understand

the interactive sign construction for the graphical sign transferred to empty

fields which is observable in this episode again as a contextual generaliza-

tion (Radford, 2003b); with writing down an arithmetical equation a first

symbolical generalization begins, even though it does not yet use algebraic

notations (cf. Radford, 2003b).

The two classroom episodes illustrate, important possibilities for a be-

ginning interactive construction of mathematical signs. In the work on the

operative structures in the number walls, the teacher repeatedly hinted at

a comparison with the initial wall. Matthi also expresses this comparison

in his deictic signs and then he transfers this showing, comparing to not

realizable changes in the transition from the first to the second row of the

wall. The sign of the comparing showing is to stand for a mathematical re-

lation and to try and record it. This function of embodying a mathematical

relation, which becomes visible in the sign production is essential for an

interactive creation of mathematical signs.

In the second episode, a further feature of interactive sign construc-

tions in the forms of generalizing transfers becomes visible. On the one

hand, a familiar sign is transferred to a new, more general situation, on

the other hand, a new sign is constructed for the same reference situation.

These forms of transfer with signs in general are known as metaphors and

metonymies (cf. Pimm, 1992) respectively. For mathematical signs the

generalizing embodiment of mathematical relations is a specific aspect of

these metaphoric and metonymic transfers.


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HEINZ STEINBRING

Institut fur Didaktik der MathematikUniversitat DortmundVogelpothsweg 87Dortmund 44221GermanyTelephone: 49 231 755 2176Fax: 49 231 755 2948E-mail: [email protected]

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