Classroom Discourse in Mathematics: A Multisemiotic Analysis

Classroom Discourse inMathematics: A Multisemiotic

Analysis

Kay L. O'HalloranNational University of Singapore, Singapore, Singapore

The impact of the multisemiotic nature of mathematics on classroom discourse is

examined from a systemic functional linguistic perspective. Mathematical discourse is

multisemiotic because it involves the use of the semiotic resources of mathematical

symbolism, visual display and language. There is constant movement between the

three resources as the primary code and with shifts between spoken and written modes

in classrooms. The complexity of pedagogical discourse arises because each of the

three semiotic resources have their own unique lexicogrammatical systems for

encoding meaning, and these interact to shape the nature of constructions found in the

classroom. This is especially significant in the case of mathematical symbolism where

meaning is encoded unambiguously in the most economical manner possible through

specific grammatical strategies, one of which involves the use of multiple levels of

rankshifted configurations of mathematical Operative processes and participants. The

dense texture of mathematics pedagogical discourse arises from the inclusion of these

symbolic constructions in the linguistic metadiscourse. The complex tracking and

reference patterns arising from the reconfigurations of symbolic Operative processes

and participants in the board text and oral discourse are also examined. Lastly, the

notion of semiotic metaphor whereby shifts in meaning of functional element occur and

new entities are introduced with movements between semiotic codes is discussed.

INTRODUCTION

Interest in mathematical language and its implications for the teaching and learn-

ing of mathematics has steadily grown over the last decade (Cocking & Mestre,

1988; Elliot, 1996; Mousley & Marks, 1991; Pimm, 1987 for example). As Mor-

gan (1996) claims, however, the focus of those investigations has largely centered

around vocabulary, symbolism and isolated examples of specialist grammatical

Direct all correspondence to: Kay L. O'Halloran, National University of Singapore, Department of

English Language and Literature, Block AS5, 7 Arts Link, Singapore 117570. E-mail:

[email protected]

Linguistics and Education 10(3): 359±388. All rights of reproduction in any form reserved.

Copyright D 2000 by Elsevier Science Inc. ISSN: 0898±5898

forms. This suggests that what has been lacking to date within the community of

mathematicians and mathematics educators is a coherent linguistic theory. Given

the traditional compartmentalization of academic disciplinary fields, this is not so

surprising. M.A.K. Halliday's systemic functional model of language, however,

provides a comprehensive linguistic theory from which to proceed.

The problem of investigating `̀ mathematical language,'' however, extends be-

yond the examination of particular linguistic selections that occur in ma-

thematical texts and classroom discourse. Rather, a critical reading of any form

of mathematical discourse must necessarily take into account the multisemiotic

nature of its makeup. Mathematics is not construed solely through linguistic

means. Rather, mathematics is construed through the use of the semiotic re-

sources of mathematical symbolism, visual display in the form of graphs and dia-

grams, and language. In both written mathematical texts and classroom

discourse, these codes alternate as the primary resource for meaning, and also

interact with each other to construct meaning. Thus, the analysis of ``ma-

thematical language'' must undertaken within the context of which it occurs;

that is, in relation to its codeployment with mathematical symbolism and visual

display. The analysis of the language of mathematics classrooms must necessarily

be incomplete unless the contributions and interaction of the symbolism and vi-

sual display are taken into account.

Significantly, the Hallidayan model provides a theoretical framework for the

analysis of semiotic systems other than language as demonstrated by, for exam-

ple, O'Toole's (1994; 1995) systemic model for the visual arts. Thus systemic

functional theory can be extended to investigate the lexicogrammar of ma-

thematical symbolism and visual display. Although not presented here, preli-

minary frameworks which outline the major systems for these resources

have been constructed and used for the analysis of the oral discourse and board

texts in mathematics classrooms (O'Halloran, 1996). Each of the cited examples

in this paper is from this larger research project.

As comprehensive descriptions of systemic functional grammar of the English

language (Halliday, 1994; Martin, 1992; Matthiessen, 1995) are available, to-

gether with introductory texts (Butt, Fahey, Spinks, & Yallop, 1996; Eggins,

1994; Martin, Matthiessen, & Painter, 1997; Thompson, 1996) I limit my discus-

sion to the extensions of the model, which involve mathematical symbolism and

visual display.

THE MULTISEMIOTIC NATURE OF MATHEMATICAL DISCOURSE

Mathematics is multisemiotic because the linguistic, visual and symbolic semiotic

systems differentially contribute to the meaning of the text. Following Lemke

(1998), each semiotic system is basically uniquely functional in its contribution

in the construction of mathematical meaning. I briefly consider the functions of

360 O'HALLORAN

mathematical symbolism, visual display and natural language before considering

the interaction between these semiotic systems. Here, the emphasis is directed to-

wards what is perceived as most problematic, the interaction between the re-

sources of mathematical symbolism and language.

The Functions of Mathematical Symbolism, Visual Display, and Language

Mathematical symbolism has evolved historically to fulfill particular func-

tions that are not possible with other semiotic resources. As Lemke (1998) ex-

plains, semantically mathematical symbolism exceeds the potential of language

as it allows for topological descriptions of continuous patterns of covariation.

This may be compared to the potential of natural language that primarily realizes

typographical or categorical descriptions. For example, Burgmeier, Boisen, and

Larsen (1990, p. 83) give the mathematical description s(t) = ÿ16t 2 + 80t for the

height of an arrow shot vertically into the air, where t is the time in seconds. In

this mathematical symbolic description, the complete pattern of the relationship

between time and height of the arrow is encoded. However, using the semiotic

resource of language, we could only say, for instance, `̀ the arrow is still rising,''

or `̀ it is falling'' or that `̀ it has hit the ground.'' The mathematical description

gives the exact description of the height of the arrow at any point in time, a feat

not possible with any other semiotic resource.

The visual semiotic in mathematics is also important for it allows perceptual

understanding of the relationship between entities which is encoded symbolically.

Returning to the previous example, the relationship between time and the height

of the arrow encoded in s(t) = ÿ16t 2 + 80t is displayed visually in Fig. 1. The

graph gives the viewer insight into the nature of this relationship that corresponds

to our sensory experience of the world. The visual semiotic thus uniquely contri-

butes to meaning in mathematics where the overall pattern of a relationship may

be gleaned at a glance.

There is much room for investigation of visual representation in mathematics.

Preliminary analyses indicate that the visual semiotic is critical at particular stages

in the solution of mathematical problems, and further to this, the specific lexico-

grammar of visual representation needs to made explicit for a complete interpreta-

tion of the text (O'Halloran, 1996). However, for purposes of this discussion, the

issues I address largely concern mathematical symbolism. The reason for this

choice is the considerable impact on the nature of classroom discourse that arises

from the use of this semiotic. Therefore, I first discuss lexicogrammatical strate-

gies in mathematical symbolism that allow the descriptions of continuous varia-

tion to be possible. Following this, I examine how these selections impinge on

classroom discourse. This is perhaps best approached through historical contex-

tualization of the evolution of mathematical symbolism, for here lies the basis

for the nature of contemporary symbolic forms.

CLASSROOM DISCOURSE IN MATHEMATICS: A MULTISEMIOTIC ANALYSIS 361

The Evolution of the Lexicogrammar of Mathematical Symbolism

A historical look at the evolution of the genres of mathematical texts suggests

that the lexicogrammar of mathematical symbolism may have evolved from na-

tural language because mathematical texts were initially written in the prose form

of verbal `̀ rhetorical algebra.'' These texts contained detailed verbal instructions

about what was to be done for the solution of a problem. In later texts, there ap-

peared abbreviations for recurring participants and operations in what is known

`̀ syncopated algebra.'' The use of variables and signs for participants and ma-

thematical operations in the last 500 years resulted in `̀ symbolic algebra'' and

the contemporary lexicogrammar of mathematics. Thus, we may conjecture that

the grammar of modern mathematical symbolism grew directly out of the lexico-

grammar of natural language and this may explain the high level of integration of

symbolic and linguistic forms in mathematical texts.

It appears that only circumscribed ranges of options from the meaning poten-

tial of natural language were selected for encoding in mathematical symbolism.

For example, with respect to experiential meaning, the full range of options for

process selections in language (that is, material, relational, verbal, mental, beha-

vioral, and existential processes) are not symbolized in mathematics. Rather, ma-

thematical symbolism is concerned with a narrow range of process types. The

contraction of options meant that maximal structural condensation with unam-

biguous meaning could occur in the symbolism while simultaneously allowing

semantic extensions that exceeded the meaning potential of language. As I dis-

cuss in Operative Processes in Mathematical Symbolism, one mechanism

through which semantic extensions took place in mathematics was the develop-

ment of a new process type. However, as a consequence of the nature of its evo-

lution from natural language and the accompanying contraction and expansion of

particular realms of meaning, mathematical symbolism never gained an overall

Figure 1. Graph for s( t ) = ÿ16t 2 + 80t.

362 O'HALLORAN

functionality as a semiotic system. Mathematical symbolism thus requires code-

ployment with language with the former acting as the contextual metalanguage

of mathematics.

This supporting role of natural language in mathematics means that semantic

extensions must have also taken place in this semiotic to incorporate the expan-

sion of meaning potential permitted by the mathematical symbolism. Here, I shall

attempt to recontextualise the evolution of grammatical metaphor in science (Hal-

liday & Martin, 1993; Martin, 1997; Martin & Veel, 1998) in terms of its func-

tionality in incorporating the semantic extensions provided by mathematical

symbolism. The interaction between the two semiotic systems of mathematical

symbolism and language has also resulted in further semantic extensions that

would not have otherwise been possible. These arguments are further developed

in Grammatical Metaphor.

From another perspective, mathematical symbolism may also be viewed as

evolving to bridge the gap between linguistic descriptions and perceptual reality

(O'Halloran, 1996; 1999a; 1999b). As previously discussed, linguistic descrip-

tions are largely limited to categorical type classifications. On the other hand, vi-

sual descriptions are partial representations of continuity that are linked to

perceptual reality. Historically, the link from text-based descriptions to visual re-

presentations was perhaps finally achieved with the development of Cartesian

geometry and calculus where the `̀ grammatical metaphor,'' or the mathematical

symbolism with its origins in language, was linked to the `̀ visual metaphor,'' the

abstract diagrams and graphs. Before this time, algebra appeared to be in a state

of disarray (Kline, 1972). However, the linking of text-based descriptions in the

form of symbolic statements to visual representations meant that mathematical

symbolism rapidly developed as an integrated semiotic system. This increased

functionality arising from the intimate links between linguistic, symbolic, and vi-

sual representations allowed the construal and solution of problems that were pre-

viously not conceivable.

The functions of the three semiotic resources in mathematics may thus be sum-

marized as follows: the mathematical symbolism contains a complete description

of the pattern of the relationship between entities, the visual display connects our

physiological perceptions to this reality, and the linguistic discourse functions to

provide contextual information for the situation described symbolically and vi-

sually. The major reason why mathematical symbolism is generally accorded

the highest status by mathematicians is because this is the semiotic through which

the solutions to problems are derived. In this respect, the visual display is not

only limited in functionality, but also graphs and diagrams are usually only partial

descriptions of the complete description encoded in the mathematical symbolic

statements. In addition, there exists a possible misuse of diagrams, although Shin

(1994) maintains that each of the above claims do not entirely warrant the lower

status accorded to mathematical visual display. With the power of computers to


dynamically display visual images, however, the status of this semiotic appears to

be rapidly increasing.

In Operative Processes in Mathematical Symbolism, I discuss the evolution

of the new process type, Operative processes, in mathematical symbolism.

This is a necessary preamble to the discussion of the lexicogrammatical stra-

tegies through which the semantic extensions in mathematical symbolism are

made possible.

Operative Processes in Mathematical Symbolism

Semantic extensions in mathematics perhaps first took place with the early de-

velopment of numerical systems where material processes of increasing, decreas-

ing, combining and sharing were gradually replaced with arithmetical processes

of adding, subtracting, dividing and multiplying. These symbolic numerical pro-

cesses gave results which were unexpected (for example, the multiplication of

fractions where smaller rather than larger quantities are derived) and quantities

that were previously inconceivable (for example, extremely large or small frac-

tional quantities). Operations were thus performed on new entities in ways that

were not feasible in the everyday commonsense world. I have called these

new process types Operative processes (O'Halloran, 1996; 1999b). Operative

processes may be conceived as processes performed on mathematical objects

such as numbers and later, variable and other abstract quantities. Operative pro-

cesses include the four arithmetic operations of addition, subtraction, multiplica-

tion and division and later algebraic operations such as those found in advanced

and higher mathematics.

Operative processes appear to be grammatically different from the processes

found in natural language because the notion of one Medium is perhaps no longer

feasible nor sensible. Here I am referring to Halliday's definition of a Medium as

the key participant through which the process is actualized. That is, in linguistic

construals of our experience of the world, associated with each process is a single

Medium through which the action, sensing or happening takes place. In ma-

thematical statements, this notion of one Medium may no longer be applicable.

For example, in the symbolic description of the plane, 2x + 3y + 4z = 12, and the

volume of a rectangular prism, V = l � b � h, there appears to be equal status

between the three participants, x, y and z, and l, b and h, each of which appears

to be central to the Operative processes through which the relation is encoded.

Perhaps in mathematics, we enter a new semantic realm where we are capturing

multiple interactions with an unlimited number of participants in a way that is not

found in linguistic constructions.

Most importantly, this notion of Operative processes becomes critical for the

investigation of pedagogical discourse, for it is precisely the lexicogrammatical

strategies for encoding nuclear configurations of Operative processes and partici-

364 O'HALLORAN

pants in mathematical symbolism, which appear to be the major factors contribut-

ing to difficulties in the mathematics classroom. By a nuclear configuration, I

mean a specific semantic cluster that involves an Operative process together with

the associated participants.

In what follows, I restrict the discussion of the impact of the multisemiotic

nature of mathematics on classroom discourse to three main areas: first, the na-

ture of the lexicogrammar of mathematical symbolism and its effect on the sur-

rounding verbal discourse; secondly, some general features of mathematical

pedagogical discourse arising from the multisemiotic nature of its makeup;

and, thirdly, the shifts in meaning that result with movements between codes.

With respect to the latter, the shifts between the semiotic codes of mathematical

symbolism, visual display and language give rise to the phenomenon of semio-

tic metaphor whereby shifts in the functions of individual elements occur and

new entities are introduced. As illustrated below in the analysis of the class-

room discourse, this phenomenon is similar to grammatical metaphor in lan-

guage, but occurs in shifts between codes rather than within language. One

of the inherent difficulties for mathematics students is that these shifts in mean-

ing, which occur with movements between the codes, in particular, the verbali-

zation of mathematical symbolic statements.

This discussion is not exhaustive and only represents preliminary findings

arising from the analysis of mathematics lessons in secondary schools (O'Hal-

loran, 1996). In addition, while the following observations are largely concerned

with the mathematical symbolism, as previously mentioned, this is not to sug-

gest that learning difficulties do not occur as a result of the lexicogrammar of

the visual display in mathematics. This semiotic has a unique lexicogrammar

that must be made explicit for the interpretation of visual representation, as per-

haps illustrated by Fig. 1. To interpret this graph, the lexicogrammar of the axes

and its relation to the curve depicting the mathematical relation must be under-

stood. The focus of this discussion, however, centers on the lexicogrammar of

mathematical symbolism.

THE GRAMMAR OF MATHEMATICAL SYMBOLISM

The lexicogrammar of mathematical symbolism exploits specific lexicogramma-

tical strategies that differ from those found in natural language. The following

discussion is concerned with the nature of these strategies and the difficulties

these cause in the classroom context.

Lexicogrammatical Strategies for Encoding Experiential Meaning

Mathematical symbolism operates through lexicogrammatical strategies of

structural condensation which are, in part, possible because of the restricted range


of meanings. Structural condensation, whereby meaning is encoded in the most

economical way possible, is achieved through devices such as:

1. a specific rule of order for Operative processes (brackets, indices,

multiplication/division and addition/subtraction)

2. the possibility of alternate orderings through use of different forms

of brackets

3. ellipsis of the Operative process of multiplication

4. exploitation of the resources of spatial graphology

5. conventionalized symbolic forms

For example, in the case of s(t) = ÿ16t2 + 80t, the condensatory devices that

have been employed include the rule of order so that multiplication, ÿ16 � t � t

and 80 � t, occurs before addition of these two terms, ellipsis of the multiplica-

tion sign in ÿ16t2 and 80t, and use of spatial graphology so that t2 means t � t.

Other strategies involve the use of standard functional notation; for example, s(t)

to mean the value of the function s at t. With further investigation of the lexico-

grammar of mathematical symbolism, the devices of structural condensation

could be fully documented.

One major function of the structural devices for condensation is to permit mul-

tiple levels of rankshifted nuclear configurations of Operative processes and par-

ticipants to be preserved in an economical and unambiguous form in mathematical

symbolic statements. This is best explained through a comparison with language,

which may be conceived as consisting of an ascending constituent rank scale of

word, word group/phrase, clause and clause complex or sentence (Halliday,

1994, p. 23). Rankshift, or downranking, occurs when one rank (for example

the clause) operates at a lower rank (for example, the word group/phrase). That

is, a functional element within a clause may be another clause. In a similar manner,

by considering the previous example s(t) =ÿ16t2 + 80t in connection with the lin-

guistic rank scale, we may see that mathematical symbolism can be conceived as

consisting of an ascending rankscale of:

. term or `̀ atom'': s(t), 16, t, 80, + (i.e. word)

. expression: ÿ16t2, 80t, ÿ16t2 + 80t (i.e., word group/phrase)

. clause: s(t) = ÿ16t2 + 80t (i.e., clause)

. clause complex: s(t) = ÿ16t2 + 80t ; s(t) = ÿ16t (t ÿ 5) (i.e.,

clause complex)

Here, the term or atom, expression, clause and clause complex in mathematical

symbolism is analogous to the rank of word, word group/phrase, clause and clause

complex, respectively, in language. However, we may see that in the example

above, Operative process and participant configurations occur at the rank of ex-

366 O'HALLORAN

pression. That is, expressions in mathematical statements (for example,ÿ16t2, 80t

and ÿ16t2 + 80t) consist of rankshifted clauses, or configurations of Operative

processes and participants (ÿ16 � t � t, 80 � t and ÿ16 � t � t + 80 � t). The

exact nature of these rankshifted nuclear configurations may be demonstrated

through closer examination of s(t) = ÿ16t2 + 80t.

As displayed Table 1, this statement involves a Relational process with par-

ticipants s(t) and ÿ16t2 + 80t. The latter participant, ÿ16t2 + 80t, consists of a

rankshifted clausal configuration of the Operative process of addition and two

expressions ÿ16t2 and 80t. Furthermore, as shown in Table 1, these two expres-

sions, ÿ16t2 and 80t, are also rankshifted clausal configurations of atoms/ex-

pressions and Operative processes; that is, ÿ16 multiplied by t 2 and 80

multiplied by t. In turn, the expression t2 is a rankshifted clausal configuration

of t multiplied by itself. The extent of the depth of the clausal rankshift becomes

clear in the following bracketing of the configurations of Operative processes

and participants:

s�t� � ��ÿ16��t 2�� 80t�� for s�t� � ÿ16� t � t � 80� t:

The grammatical strategy of clausal rankshift that functions to preserve nu-

clear configurations of Operative processes and participants in mathematical

symbolic statements thus appears to be more sophisticated than that found in

language. While the grammar of language also allows experiential meaning to

be packed through the process of rankshift, where a clausal element or phrasal

element functions at a lower rank, as for example, `̀ the solution [[we found]] is

incorrect,'' or `̀ the solution [in the book] is wrong,'' mathematical symbolism

appears to allow for a greater depth of clausal rankshift. In natural language ex-

periential meaning tends to be packed or condensed through alternative strate-

gies that include nominalization and the formation of extended word groups.

Table 1. Rankshifted Nuclear Conf igurations of Mathematical Processes and Partici-

pants in s(t ) = ÿ16t 2 + 80t

Rank Process Participants

Rank 1 (Ranking clause) = (Relational) . s(t)

. [[[[ÿ16[[t 2]]]] + [[80t]]]]

Rank 2 + (Operative) . [[ÿ16[[t 2]]]]. [[80t]]

Rank 3 (i) � (Operative) . ÿ16. [[t 2]]

Rank 3 (ii) � (Operative) . 80

. t

Rank 4 � (Operative) . t

. t


These grammatical strategies for condensation in language mean that process/

participant nuclear configurations are lost. However, in mathematical symbo-

lism, the Operative processes between mathematical participants, which encode

the mathematical description, are left intact, unlike strategies in language, which

remove process and participant configurations. I return to this point in the dis-

cussion of grammatical metaphor in language in Grammatical Metaphor. The

preservation of configurations of Operative processes and participants in ma-

thematical symbolic statements is critical, for here lies the means through which

relations between participants are encoded and rearranged for the solution to

mathematical problems.

The Lexicogrammar of Mathematical Symbolism and Classroom Discourse

The analyses of discourses in mathematics classrooms (O'Halloran, 1996)

suggest that the degree of rankshifted configurations of Operative processes

and participants in the mathematical symbolism causes difficulties for students

with respect to accessing the experiential meaning of the board text. Further to

this, the differing lexicogrammatical strategies in symbolic mathematics impinge

on the nature of constructions found in the oral discourse. To examine these

claims in more detail, we may consider the section of a board text displayed in

Fig. 2. This text arose in a trigonometry lesson in Perth, Western Australia in a

Year 10 class with 15-year-old students. All the cited examples in trigonometry

arose in the oral discourse and board texts of this lesson.

At the discourse semantics stratum (Martin, 1992), the basic nuclear relations

for the section of this board text are displayed in Table 2. Here, nuclear relations

are construed as a description of the center (the Process), the nucleus (the Med-

ium) and the margin (the Agent and Beneficiary).

As displayed in Table 2, we may see that there are up to three levels of

rankshift where participants in the ranking clause are themselves the product

of nuclear configurations of Operative processes and participants. In many

cases, the embedded Operative process of multiplication is implicitly realized

through the grammar of mathematical symbolism that, in some cases, involves

the use of brackets. The Operative process of division is realized by spatial

positioning and use of the fraction line. At the deepest level of embedding,

Figure 2. Section of Board Text: Trigonometry Lesson.

368 O'HALLORAN

the participants are r, h, 10, tana and tanu. We may conjecture from Table 2 that

another potentially problematic situation for students involves reorganizing these

participants and Operative Processes and expressing the new configuration ac-

cording to the lexicogrammar of mathematical symbolism. This phenomenon

is further explored with respect to reference patterns for textual meaning in

The Lexicogrammar of Mathematical Symbolism and Classroom Discourse.

However, at this stage, I would like to focus on part of the oral discourse that

precedes the section of the board text. The part of the oral discourse in question

is displayed in Table 3.

There are two discernible features of the nuclear relations of the oral dis-

course accompanying the board text as we may see from Table 3. First, the stu-

dent or the teacher may be the major participant in the oral discourse. That is,

the Agent and Medium are frequently either the teacher or the students. Sec-

ondly, there is a range of processes that constitute the Center. This may be

compared to corresponding symbolic text displayed in Table 2 where the tea-

Table 2. Nuclear Relations in the Board Text

Center Nucleus Margin

Clause

PROCESS

= Range:

process

+ Medium

+ Range: entity

+ Agent

+ Beneficiary

10 = [[[[r (tanu)]]ÿ[[r (tana)]]]]

Rank 1 = . [[[[r (tanu)]]ÿ[[r (tana)]]]]

. 10

Rank 2 ÿ . [[r (tanu)]]. [[r (tana)]]]

Rank 3(i) � . ``r''

. tanuRank 3(ii) � . ``r''

. tana10 = [[r [[(tanu ÿ tana)]]]]

Rank 1 = . [[r [[(tanuÿtana)]]]]

. 10

Rank 2 � . ``r''

. [[(tanu ÿ tana)]]Rank 3 ÿ . tanu

. tanar = [[(10)/([[(tanu ÿ tana)]])]]

Rank 1 = . [[(10)/([[(tanuÿ tana)]])]]

. ``r''

Rank 2 � . 10

. [[(tanuÿtan a)]]Rank 3 ÿ . tanu

. tana


cher and students as Agents are ellipsed and the processes are restricted to

Relational and Operative processes. This indicates that differing nuclear con-

figurations of the ranking clauses are found in the oral and symbolic texts.

This is not surprising, given the differing functions of each code. That is, the

solution of the problem is derived symbolically and the oral discourse tends to

function as the metadiscourse for this procedure. However, although the spoken

discourse is indispensable for contextualizing the actions that are performed,

and the result which is obtained symbolically, the levels of embedding for

the symbolic participants and processes in the oral discourse are nevertheless

increased with inclusion of teacher and student participants. For example, there

are examples in the oral discourse where the entire mathematical statement is

rankshifted, for example:

Student right well you go [[[[`̀ r'' tan of theta take `̀ r'' tan of alpha]]equals ten]]

With the inclusion of the teacher and students, the degree of rankshift in the

oral discourse is at times greater than that occurring in the symbolic discourse.

Even when the teacher and students are not realized as participants, the packing

Table 3. Nuclear Relations in the Oral Discourse Preceding the Board Text

Center Nucleus Margin Periphery

Clause

PROCESS

=Range:

process

+Medium

+Range: entity

+Agent

+Beneficiary

�Circum-

stance

Teacher // John

Student // ahh since

hsincei you know

like tan of hhyou

knowii theta

/umm/ theta and

alpha

know . you

. like tan of

hhyou knowiitheta /umm/

theta and

alpha

Student // right well you go

[[[[`̀ r'' tan of theta

take `̀ r'' tan of

alpha]] equals ten]]

go . [[[[``r'' tan of

theta take ``r''

tan of alpha]]

equals ten]]

. you

Student // OK so therefore

[[`̀ r'' [[tan of theta

take tan of alpha]]]]

equals ten

equals . ten . [[``r'' [[tan of

theta take

tan of

alpha]]]]

Student // and `̀ r'' equals

[[ten on [[tan of

theta take tan of

alpha]]]]

equals . [[ten on [[tan

of theta take

tan of alpha]]]]

. ``r''

Teacher // OK well done

370 O'HALLORAN

of the nuclear configurations is nonetheless dense. While devices for simplifica-

tion of the complexity of the lexical items in the oral discourse include, for ex-

ample, exophoric references such as `̀ that'' for clauses and participants in the

symbolic text, students still nonetheless encounter multiple levels of clausal rank-

shift in mathematics classroom discourse.

One difficulty in penetrating mathematical discourse is the degree of clausal

rankshift that occurs in both the oral discourse and the symbolic text. The parti-

cipants in the ranking clause are the result of multiple levels of rankshifted con-

figurations of mathematical processes and participants. The levels of nuclear

relations that form each of the participants in the ranking and non-ranking clauses

must be unpacked. However, a second difficulty occurs with the reconfiguration

of these nuclear configurations as participants in the following ranking clause.

Further to this, as discussed in Logical Meaning in Mathematics Discourse, the

logical basis on which operations are performed and nuclear relations configured

or reorganized are implicit mathematical results arising from definitions, proper-

ties and laws. So, not only is the action embedded and submerged in ma-

thematical discourse, it is also based on implicit results. In The Reconfiguration

of Operative Processes and Participants, I examine the problems involved in track-

ing and reconfiguring mathematical participants before examining this aspect of

logical meaning in mathematics.

The Reconfiguration of Operative Processes and Participants

The tracking of mathematical participants in reconfigurations of Operative

processes as conceptualized through the system of IDENTIFICATION (Martin,

1992, pp. 93±157) is complex in mathematics classrooms. An example of the re-

ference patterns for a segment of the trigonometry lesson is given in Fig. 3. Apart

from the teacher and the students, the major participants in the oral discourse are

the participants in the symbolic text; h, r, 10, tana and tanu. Here the complexity

of the tracking patterns across the oral and written discourse is highlighted, and

we may also gain some idea of the extent of reconfigurations that occur.

The lexical chains and strings (Martin, 1992, pp. 271±379) for mathematical

participants and processes are in a constant state of splitting and cojoining. This is

complex, as the lexical relations realized in the oral discourse in mathematics

classrooms are intertwined with those in the written symbolic and visual texts.

The cohesion of the oral discourse is thus often dependent on contextual factors.

However, one major problem here for students is the reorganization of the new

configurations based on mathematical results and definitions according to the

lexicogrammar of mathematical symbolism. In order to examine this in more

detail, I shall refer to Fig. 4, where the reference chains for the board text are dis-

played. The rectangular boxes in Fig. 4 indicate participants that, in many cases,

are rankshifted nuclear configurations of Operative processes and participants.


Fig

ure

3a

.M

ajo

rR

efer

ence

Ch

ain

sin

the

Ora

lD

isco

urs

e.

372 O'HALLORAN

Fig

ure

3b

.(C

on

tin

ued

)


Fig

ure

3c.

(Co

nti

nu

ed)

374 O'HALLORAN

The symbolic solution to the trigonometric problem involves recombining the

participants through mathematical Operative processes into new nuclear config-

urations. Initially distinct participants become blended into single participants and

composite participants are split into separate participants and then recombined

with others. For example, h and 10 combine to become h-10. This participant

is split into h and 10 + r (tana) which is then reconfigured on several more occa-

sions. Participants thus intertwine and combine according to Operative processes

that are not always explicitly marked in the symbolic text. Difficulties may arise

in actually identifying the different nuclear configurations. What serves to exacer-

Figure 4. Major Reference Chains in Symbolic Written Text.


bate the problem is that these participants split and cojoin in what are largely

rankshifted nuclear configurations.

The reference chains of the board text displayed in Fig. 4 bear further scru-

tiny. As reflected in the oral discourse, the reference chains are intricate through

patterns of cojoining and splitting. In the symbolic written text, this often re-

sults in spatial crisscrossing of chains. The economy of the lexicogrammar of

mathematical symbolism contributes to the density of these reference chains.

The spatial positioning of the mathematical statements, however, realizes tex-

tual meaning. Mathematical texts are highly conventionalized with strategies

that include the spatial arrangement of sequential mathematical statements

and the ordering of participant functions. The notion of a Theme as the point

of departure of a message and Rheme as the part in which the Theme is devel-

oped (Halliday, 1994, Chapter 3) is highly organized to the point of being gen-

eric. In Fig. 4, each consecutive statement appears on a new line spatially

positioned directly underneath the previous one as an aid for the tracking of

participants. Despite this strategy, however, as can be seen by the reference pat-

terns in Fig. 4, the chains are nevertheless complex. As for the oral discourse,

tracking is complicated as the real action occurs within rankshifted nuclear con-

figurations of Operative processes and participants. Further to this, there are

multiple levels of rankshift as indicated by the rectangular boxes in Fig. 4. That

is, there is no problem in identifying the participants in the ranking clauses, but

these participants must be unpacked in order to understand how they relate to

the previous nuclear configurations.

To alleviate difficulties in constructing the appropriate reference chains, the

teacher uses two strategies in the board text as aids for tracking. First, the tea-

cher labels the two mathematical statements `̀ (1)'' and `̀ (2)'' and then uses

these labels explicitly as an aid for reference. Secondly, the teacher uses head-

ings realizing commands `̀ Algebraically obtain ONE variable as the subject of

EACH equation,'' `̀ Equate expressions for h'': and `̀ Substitute r into eq_n (2)'':

Although this text is a running commentary on the procedures that are per-

formed to arrive at each result, it also aids the tracking of the participants.

The two strategies of explicit labelling and linguistic commentary compensate

for the lack of selections from the system of DEIXIS. However, it would appear

near impossible to track or even identify participants without prior knowledge

of the grammar of mathematical symbolism and the mathematical results that

have been used to combine or split the embedded participants. I discuss this

basis for logical meaning in mathematical discourse in Logical Meaning in Ma-

thematical Discourse.

In addition, as for scientific discourse (Halliday, 1993, pp. 72±74), highly or-

ganized taxonomic relations and interlocking definitions may cause difficulties in

mathematical discourse. In the analysis of the trigonometry lesson, for example,

taxonomic relations play an important role in the cohesion of the oral discourse.

376 O'HALLORAN

The mathematical register items are related to each other through class, similarity

or through compositional relations with respect to, in this case, the triangles, ver-

tices, angles, sides and the relations between these parts of the triangle as ex-

pressed through the trigonometric ratios. The taxonomic relations also extend

to algebraic terms such as `̀ equation,'' `̀ expression,'' `̀ statement,'' and `̀ subject''

in addition to the lexical choices for Operative processes. Here we may note that

the range of taxonomic relations relate to options within the systems for visual

display in the form of the diagram and as well as the mathematical symbolism.

Students must have knowledge of these taxonomies in order to follow the sequen-

tial steps in the solution to mathematical problems.

GENERAL FEATURES OF MATHEMATICAL

PEDAGOGICAL DISCOURSE

In this section, I briefly examine some general features of the impact of multise-

miotic nature of mathematics on pedagogical discourse and other complexities

that arise with respect to logical, experiential and interpersonal meaning.

Logical Meaning in Mathematics Discourse

The logical basis for mathematics may be not entirely deductive in nature, but

rather a collection of algorithmic systems and derivable sentences generated from

a postulate base (Azzouni, 1994; Tiles, 1991 for example). From this perspective,

the explicit statement of the algorithmic steps involved in the derivation of solu-

tions gradually became implicit as the grammar of the symbolism developed and

the results became generalized. The algorithms thus became implicit procedures

using established results derived from a recursive set of axioms and inference

rules. The implicitness of the deductive and operative relations in generic steps

in the mathematical symbolic texts may cause problems in the classroom since

very often there are long chains of reasoning that provide little or no indication

of the results, definitions, axioms, operational properties or laws that have been

used. The complexity increases with the rankshifted nature of the mathematical

reconfigurations under consideration.

The analysis of the oral discourse in mathematics lessons reveals a high inci-

dence of consequential type relations that result in long implication chains of rea-

soning (O'Halloran, 1996). As language functions as metadiscourse for the

symbolic problem, the basis for the multiple occurrences of consequential rela-

tions through which the chains of reasoning are constructed are located within

the written mathematics text. For example, the logical relations for the board text

of the trigonometry lesson are displayed in Fig. 5.

From Fig. 5, several features of the logical relations become apparent. First

and foremost, the relations are internal and organize mathematical semiotic rea-


lity as opposed to material everyday reality. The relations organize a rhetorical

argument, and the text as such is genre-structured as opposed to commonsense

field-structured reality. Next, apart from the symbol ; in Lines 5 and 11 that rea-

lizes cohesive logical relations of `̀ therefore'' (Martin, 1992, p. 179), the logical

relations remain implicit. The category of the implicit relations in the ma-

thematical text are largely consequential condition relations realized by `̀ so.''

Difficulties arise, however, not because the logical relations are implicit, but

Figure 5. Logical Relations in the Symbolic Text.

378 O'HALLORAN

rather because the conditions through which the logical relations are made are im-

plicit. Generically, mathematical symbolic discourse realizes consequential logi-

cal relations, so the rhetorical structure comes as no surprise. What does vary is

the postulate basis or the properties, definitions and results through which these

logical connections are made. For example, the mathematical definitions and

properties that have been assumed in the consequential relations in Fig. 5 include

the definition of the tangent ratio1(lines 1 and 2), the Multiplication Property of

Equality2 (lines 4, 6, 11), the definition of the Multiplicative Inverse3 (lines 4, 6,

11), the Addition Property of Equality4 (lines 5 and 9), the Definition of Subtrac-

tion5 (lines 5 and 9), the Equality Property6 (lines 8 and 13) and the Distributive

Property of Multiplication Over Addition7 (line 10).

The nature of the logical relations highlights the need for unpacking the ex-

periential meaning realized by the grammar of mathematical symbolism together

with the mathematical definitions and properties on which the logical relations

are based. These conditions are instrumental in constructing arguments realized

through mathematical symbolism. Otherwise, students are required to make ma-

jor semantic leaps (Halliday, 1993, pp. 82±83). As Halliday states `̀ . . . writers

[of scientific discourse] sometimes make semantic leaps, across which the reader

is expected to follow in order to reach a required conclusion'' (p. 83). It appears

that mathematical discourse is full of such semantic leaps.

Interpersonal Meaning in Mathematics

Using Martin's (1992, p. 529) indicators of status, mathematics is positioned

as a discourse of power. In other words, selections from the systems for inter-

personal meaning, such as the patterns of maximal modalization and modula-

tion, lack of ellipsis apart from generic conventionalized forms, the structure

of the exchanges as predominantly a series of statements or imperative com-

mands, and the general patterns of interpersonal congruence in the mathematical

symbolism combine to position mathematics texts as authoritative. In addition,

the contact or social distance is close as evidenced by specialized technical vo-

cabulary, while simultaneously being curiously distant through the predomi-

nantly monologic style and formal relations. In combination with the lack of

expression of affect, mathematics texts realize dominating interpersonal relations

that function pervasively regardless of the educational context in which mathe-

matics is taught.

The dominating interpersonal stance of symbolic mathematics is mirrored in

the visual display. This stance is realized through modal selections that include

the degree of idealization, abstraction, quantification and the prominence of the

individual figures and interplay of relations. Hence, although mathematical visual

representations clearly do not correspond exactly to material reality, they never-

theless function as `̀ truth'' since the Galilean reality of science underlies visual


modality; reality based on `̀ size, shape, quantity and motion'' (Mumford, 1934,

quoted in Kress and van Leeuwen, 1990, p. 53).

The analyses of the oral pedagogical discourses of classrooms differentiated

on the basis of social class and gender demonstrate that the nature of interperso-

nal relations in the classroom vary (O'Halloran, 1996). While mirroring the dom-

inating interpersonal stance of mathematics in classroom discourse may function

to deter students from engaging with mathematical discourse, especially those

who are accustomed to adopting a deferential position in social relations, efforts

by teachers to continually compensate and downplay the dominating interperso-

nal stance of mathematics may have an adverse effect, in that the necessary scaf-

folding to mathematics discourse is not achieved. Perhaps what is needed is to

make explicit the nature of the interpersonal dimensions of the discourse. This

recognition would allow not only access and participation in mathematical dis-

course, but also later open the way for critical intervention. In an educational con-

text, this is especially significant if interpersonal meaning is viewed as the

gateway through which we explore our experience of the world (Halliday, 1975).

MOVEMENTS BETWEEN SEMIOTIC CODES

In the following sections I explore several phenomena that occur specifically with

movements between the semiotic codes of mathematical symbolism, visual dis-

play and language. Once again, this is not an exhaustive list, but represents the

results of preliminary findings.

Syntactic Ambiguity

As within scientific discourse (Halliday, 1993, pp. 77±79), syntactic ambi-

guity may occur in mathematical pedagogical discourse. In this case, however,

the ambiguity may arise as a result of shifts between semiotic codes rather than

specific configurations of linguistic selections. When symbolic expressions are

verbalized, for instance, ambiguities that are not present in symbolic representa-

tions may occur. The reason for this discrepancy is the exploitation of differing

lexicogrammatical resources from within the different semiotic resources. For ex-

ample, two linguistic statements that occurred in the oral classroom discourse in

the trigonometry lesson are documented in Table 4.

As we may see, there are two different possible symbolic interpretations of

each linguistic statement. This problem arises as a direct consequence of the dif-

ferent lexicogrammar of language and symbolism. In the first example, two lexi-

cogrammatical resources of mathematical symbolism, spatial graphology and the

fraction line do not readily transfer to language, and vice versa. A similar situa-

tion is demonstrated in the second example where the grammar of mathematical

symbolism allows for ellipsis of the Operative process of multiplication together

380 O'HALLORAN

with brackets to indicate the grouping of nuclear configurations. When verba-

lized, however, there is ambiguity to the meaning of this linguistic statement.

In general, however, the context provides the necessary information so that

these types of ambiguities do not cause major problems in mathematical peda-

gogical discourse. The spoken discourse is accompanied by written symbolic

statements so that any misinterpretation is quickly remedied. Nonetheless, this

situation is an indication that movements between semiotic codes may cause

semantic shifts. In what follows, I discuss the notion of semiotic metaphor

where representations across semiotic codes result in semantic shifts that are

not always apparent. Before doing so, I discuss grammatical metaphor in ma-

thematical discourse.

Grammatical Metaphor

For Halliday (1994), grammatical metaphor is `̀ variation in the expression of a

given meaning'' (p. 342). The unmarked version is the congruent realization,

while other versions realizing some transference of meaning are referred to as

metaphorical forms. The presence of grammatical metaphor therefore necessitates

more than one level of interpretation, the metaphorical or transferred meaning

and the congruent meaning. Following Derewianka (1995), Halliday (1993;

1994; 1998) and Martin (1992), grammatical metaphor is organized metafunc-

tionally, and so although there exist logical, experiential, interpersonal, and tex-

tual metaphors, I limit this discussion to experiential grammatical metaphors that

most commonly occur in the form of nominalization.

Nominalization occurs when a grammatical class or structure of process, cir-

cumstance, quality or conjunction is turned into another grammatical class, that of

a nominal group or an object. Following Halliday, nominalization is conceived as

`̀ the predominant semantic drift of grammatical metaphor in modern English''

(Martin, 1992, p. 406), which has largely resulted from changes in the English

language to realize a scientific view of the world. That is, `̀ a new variety of Eng-

lish'' was created `̀ for a new kind of knowledge'' (Halliday, 1993, p. 81) where

the main concern was to establish causal relations. As Halliday explains, the most

effective way to construct logical arguments is to establish steps within a single

Table 4. Syntactic Ambiguity in Linguistic Statements of Symbolic Mathematics

Linguistic statements Symbolic versions

and `̀ r'' equals ten on tan

of theta take tan of alpha

r = 10/tanu ÿ tana r = [[10/([[(tanu ÿ tana)]])]]

r = (10/tanu ) ÿ tana r = [[[[10/(tanu)]] ÿ tana]]

OK so therefore `̀ r'' tan

of theta take tan of alpha

r tanu ÿ tana = 10 [[[[r tanu]] ÿ tan a]] = 10

equals ten r (tanu ÿ tana) = 10 [[r[[(tanu ÿ tana)]]]] = 10


clause, with the two parts of `̀ what was established'' to `̀ what follows from it''

reified as two objects or participants in the form of nominal group structures.

These two participants can then be connected with a process in a single clause.

In addition to nominalization, the strategy of recursive modification of the no-

minal group is employed in scientific discourse in order to pack information effi-

ciently. These two lexicogrammatical devices are typical of a wide range of

written discourses in the twentieth century.

Since language functions as metadiscourse for the mathematical symbolic

descriptions, linguistic choices describe not only commonsense reality, but also

mathematical reality. Language must have, therefore, evolved to incorporate

the extensions of meaning realized by the visual display and mathematical

symbolic description. The impetus for the semantic drift in language that

has been conceptualized as `̀ the world as a place where things relate to

things'' (Martin, 1993, p. 220) may be directly related to the semantic exten-

sions made possible through mathematical symbolism and visual display. The

process of grammatical metaphor is documented as one manifestation of the

adaptive strategy of natural language (Halliday & Martin, 1993; Martin &

Veel, 1998). Perhaps the meaning potential of language developed in this

way as a consequence of the functions of mathematical symbolism where de-

scriptions preserved the dynamic aspect of relations. Language could afford to

abandon these types of configurations of process and participant structures and

adopt a more static approach as a construing a world of things which are re-

lated given the functions which were fulfilled by mathematical symbolism and

its relation to visual representation.

The analysis of mathematical pedagogical discourse indicates that nominaliza-

tion and extended nominal group structures are a feature of mathematical dis-

course. Mathematical register items are predominantly nominal groups that

often involve cases of nominalization. These may involve shifts from: process

to entity, for example, the processes `̀ equate,'' `̀ assume,'' `̀ fracture'' give rise

to `̀ equations,'' `̀ assumptions'' and `̀ fractions''; shifts to object in the Qualifier

of the nominal group, for example, `̀ angles of elevation and depression'' and `̀ as

a subject of each expression''; or shifts to objects in prepositional phrases, for ex-

ample, `̀ in expression of r.'' While grammatical metaphor functions to increase

the lexical density of the discourse in mathematics classrooms, students may in-

itially also need to unpack extended nominal expressions into congruent forms

that correspond to commonsense reality.

Semiotic Metaphor

The multisemiotic nature of mathematics and the consequent interactions be-

tween the semiotic resources of mathematical symbolism, visual display and

language means that experiential metaphor is not confined to lexical and gram-

382 O'HALLORAN

matical metaphor. The shifts in semiotic codes in mathematical discourse may

be perceived to result in the phenomenon of semiotic metaphor. Although

semiotic metaphor is conceptually related to grammatical metaphor in that shifts

in the functions of elements occur, it is distinct because the shifts in meaning

take place as a consequence of movements between lexicogrammatical systems

in the different semiotic codes. For instance, as I demonstrate below, a process

or circumstance in one semiotic modality, for example, may become a partici-

pant in another modality.

The process of semiotic metaphor through which changes in the functions of

elements occur with transitions between semiotic codes may be illustrated through

consideration of the following linguistic statement that arose in an algebra lesson

(O'Halloran, 1996): `̀ the sum of the squares of two consecutive positive even in-

tegers is 340.'' The corresponding symbolic statement is: a2 + (a + 2)2 = 340. The

experiential structure of the first nominal group in the linguistic statement is given

in Fig. 6. As may be seen, there are two levels of phrasal rankshift with the experi-

ential structure of the most deeply embedded nominal group consisting of a Nu-

merative, an Epithet, two Classifiers and a Thing.

Several features of the experiential meaning of the linguistic and symbolic

texts become apparent. First, as indicated in Fig. 6, experiential meaning is

densely packed into the nominal group `̀ the sum of the squares of two con-

secutive positive even integers.'' As for mathematics, high levels of lexical

density are a source of difficulty in written scientific discourse (Halliday,

1993, pp. 76±77). However, in the translation to symbolic text, or vice versa,

it may be seen that this is a case of semiotic metaphor whereby `̀ Things'' or

objects in language, `̀ the sum'' and `̀ the squares,'' become the Operative pro-

cesses of addition, a2 + (a + 2)2 and multiplication a2 or a � a and (a + 2)2 or

(a + 2) � (a + 2), respectively, in the symbolic text. That is, there is a change

in the function of elements in the shift between the semiotic resources of lan-

guage and mathematical symbolism.

Further to this, in the nominal group `̀ the sum of the squares of two consecutive

positive even integers,'' the Epithet and the Classifier, `̀ consecutive'' and `̀ even,''

Figure 6. The Experiential Structure of the Nominal Group.


intersect experientially to give two numbers which differ by two. This results in

the Operative nuclear configuration of a + 2, which is a clausal rankshifted parti-

cipant in the symbolic text. While a common strategy for packing experiential

meaning is nominal group structures in language, the alternative strategy in ma-

thematical symbolism involves clausal rankshift of nuclear configurations invol-

ving Operative processes. The bridge between these two strategies is perhaps

semiotic metaphor whereby semantic shifts in the functions of elements occur.

In a similar fashion, the following examples from the oral discourse in the

same lesson are also instances of semiotic metaphor whereby the linguistic no-

minal groups `̀ `a' squareds,'' `̀ the squareds'' and `̀ two `a' squareds'' realize

the Operative process of multiplication, a2 or a � a and 2a2 or 2 � a � a, when

expressed symbolically.

It could be argued that the above examples are cases of grammatical metaphor,

or nominalization, in language where linguistic resources do exist for the expres-

sion of the symbolic version in a congruent rather than metaphorical form. For

example, a congruent non-technical version for a2 + (a + 2)2 = 340 would be:

`̀ When one positive number and the number obtained by adding two to this same

number are each squared and added together the result is 340.'' This rendition

also serves to demonstrate the extended length of congruent versions that do

not depend on metaphorical forms of register specific lexis.

However, it is not always the case that congruent versions for symbolic ex-

pressions are possible as illustrated by, for example, `̀ the square root of x to

the power of three'' for��x3p

. Here the linguistic version takes the form of a no-

minal group `̀ the square root of x to the power of three.'' However, the symbolic

version for `̀ the square root'' is the Operative process of�por, more precisely in

this case, (x3)1/2 where x3 is the participant. This participant, x3, is itself a rank-

shifted configuration of the Operative process of multiplication with participant x

as demonstrated by the equivalent form x � x � x. Thus there is a semantic shift

from Thing or object (`̀ the square root'') in the verbal translation to process (�p)

in the symbolic form. This indicates that when symbolic texts are verbalized, se-

mantic shifts do occur whereby, for example, Operative processes are reconstrued

as participants in language.

Other instances of semiotic metaphor include shifts to circumstantial ele-

ments for Operative processes; for example, the Operative process of division

realized by the linguistic circumstantial element `̀ on'' as found in Table 4. In

addition, there may not even exist an element to indicate the presence of an

Teacher //and then you've got to add on the `̀ a'' squareds

Teacher //because of the brackets and the squareds

Teacher //add up the `̀ a'' squareds

Teacher //so you get two `̀ a'' squareds plus your four `̀ a''

384 O'HALLORAN

Operative process in the verbal discourse. For example, the linguistic version

xy does not contain any indication of the Operative process of multiplication.

In this case, the verbal discourse functions to construe as a Thing or an object

what is symbolically a nuclear configuration of an Operative process (multipli-

cation) with two participants (x and y). In Table 4, the linguistic selection `̀ r tan

of theta'' similarly does not realize the Operative process of multiplication be-

tween r and tanu. These examples occur as a direct result of the strategies for

encoding meaning for the Operative process of division and multiplication in

the lexicogrammar of mathematical symbolism.

The process of semiotic metaphor also occurs in situations where new enti-

ties are introduced (O'Halloran, 1996). One such example arose in the transla-

tion of a verbal trigonometric problem to diagrammatic representation in the

trigonometry lesson discussed earlier. In this lesson, the trigonometric problem

involving a river and a cliff face were depicted visually by horizontal and ver-

tical line segments which were at right angles to one another. In the next step of

constructing the visual representation of the problem, the Material process of

`̀ looking down'' in the oral linguistic discourse became a participant or Thing

in the form of a hypotenuse in the diagram. The inclusion of this line segment

meant that a triangle was introduced visually. That is, in the shift from the lin-

guistically realized problem to the visual representation of the situation, the new

entity of a triangle, which previously did not exist in the oral discourse, was

introduced. This allowed the symbolic solution of the problem though the intro-

duction of trigonometric ratio for the relationship between the angles in the tri-

angle and the sides. Thus, the semantic extensions which occur with shifts to

other semiotic codes means that, in this instance, the mathematical solution to

the trigonometric problem could be derived.

As a final example, `̀ the difference between the two tangent values'' is rea-

lized symbolically as `̀ tanu ÿ tana'' in the following extract from the same tri-

gonometry lesson.

Although `̀ the difference'' is typically conceptualized as a case of grammatical

metaphor in the form of a nominalization of the process `̀ differ,'' this may also be

perceived as a case of semiotic metaphor. That is, by viewing the nominal group

`̀ the difference'' in relation to the context in which it arose, that is, in relation to

the mathematical symbolic statement, we may conceptualize this as a case of

semiotic metaphor. The Operative process of subtraction has been reconstrued

as `̀ the difference.'' Once again, it may be necessary to rethink the notion of

Teacher //so therefore you will now get an expression [[as ten divided by the difference

between the two tangent values]]

Board Text ;r = 10/(tanu ÿ tana)


grammatical metaphor and examine the context in which these forms first oc-

curred. As suggested earlier, perhaps the impetus for such shifts in meaning in lan-

guage arise from the semantic extensions provided by mathematical symbolism.

The shifts in meaning occurring in semiotic metaphor may account for learn-

ing difficulties in mathematics. That is, the semantic shifts in meaning which oc-

cur with movements between semiotic codes in mathematics are perhaps not

always recognized, especially those which occur as a result of the role of lan-

guage as metadiscourse for the mathematical symbolic statements. Students con-

strue linguistically functional elements in the symbolism as Things when in fact

they are processes or nuclear configurations of Operative process/participant

structures in the mathematical symbolism. Once again, this highlights the need

for investigation of the grammar of mathematical symbolism and the nature of

the shifts of meaning, which occur with movements between codes. Further to

this, the phenomenon of semiotic metaphor may also provide the means through

which the contributions and limitations of different semiotic codes in multisemio-

tic texts may be more fully appreciated.

CONCLUSIONS

The investigation of the multisemiotic nature of mathematics may shed new light

on difficulties inherent in the teaching and learning of mathematics. This project

would involve documentation of the unique lexicogrammatical systems that are

specific to symbolism and visual display, investigation of the functions of these

two resources and language in construing mathematical reality and exploration of

the nature of the interactions between the codes. If the meaning and thus limita-

tions of mathematics could be made explicit from a social semiotic perspective, a

truly critical reading of mathematics may become possible.

NOTES

1. The tangent ratio: In a right-angled triangle the ratio of the length of the side opposite the

given angle to that of the adjacent side.

2. The Multiplication Property of Equality: If a = b, then a � c = b � c.

3. The definition of the Multiplicative Inverse: a � (1/a) = 1.

4. The Addition Property of Equality: If a = b then a + c = b + c.

5. The Definition of Subtraction: a ÿ b = a + (ÿb).

6. The Equality Property: a = b and b = c, then a = c.

7. The Distributive Property of Multiplication Over Addition: a(b + c) = ab + ac.

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Classroom Discourse in Mathematics: A Multisemiotic Analysis