Intertextuality in school mathematics: The case of functions

LINGUISTICS

AND EDUCATION

7, 243-262

(I 995)

Intertextuality in School Mathematics: The Case of Functions ANNE CHAPMAN Murdoch University This article identifies the characteristic principles of intertextuality in school mathematics. Drawing on a larger study of the spoken language practices of the mathematics classroom, it examines how language is used to construct the shared meanings of a mathematical theme. An analysis is made of a variety of spoken and written texts in order to see how they contribute to the development of a thematic formation for the topic of functions. A pattern of thematic relations is demonstrated for each text, illustrating that each pattern is in fact a different expression of ot least part of a common pattern. The analysis identifies the text-connecting practices of language that are necessary to construct these patterns. It is argued that these practices are fundamental to the construction of mathematical meanings. The analysis also shows kinds of linguistic information that are likely to be helpful to students as they learn intertextually in mathematics.

This article is one outcome of a larger study by Chapman (1992) of spoken language practices in school mathematics. Chapman’s study was an investigation of the relations between language and learning in mathematics education. Its central premise was that mathematics education is a social practice; it constitutes a series of social events in which teachers and learners interact, primarily through spoken language, to construct and share mathematical meanings. Moreover, language is a fundamental part of the culture of school mathematics. It cannot be understood outside of the social context in which it is used. The theoretical perspective of the study was that meanings are always made in the processes of social interaction. It therefore addressed the broad question of how meanings are made. This article is narrower in scope. It examines how language is used to construct the shared meanings of a particular mathematical theme. The semantic and thematic patterns and the genre structures of school subjects are built up over the production of many texts. Moreover, every subject has its own ways of tying these texts together; its own principles of general intertextuality that determines “which things, said when. and by whom are more or less relevant” (Lemke, 1988, p. 92). My concern here is to identify, as part of the language practices of school mathematics, its characteristic principles of intertextuality. Specifically, how do students make sense of the texts they produce and encounter? What are the intertextual strategies they use to construct mathematical themes? How do texts contextualize each other? These questions are central Correspondence and requests for reprints should be sent to Anne Chapman, School of Education, Murdoch University, South Street, Murdoch, Western Australia 6150, Australia. 243

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to an understanding of how the thematic content of school mathematics is developed. The analyses to follow are of relatively small samples of the large amount of data that was collected in Chapman’s (1992) major study. The aim is to construct both a descriptive and interpretive account of the language practices of a school mathematics classroom. General intertextuality refers to the relation between the thematics or content matter of different texts. It concerns the ways in which patterns of meaning relations about a mathematical theme are created and built up across a range of seemingly disparate texts. I use this principle here as an analytic tool and also to argue that learning school mathematics involves certain kinds of text-connecting practices; that is, it involves recognizing and constructing relations among various texts. The aim is to identify these practices and show how they work. This account of general intertextuality derives from a social semiotic perspective and is necessarily underpinned by the notion of text. Text, according to Halliday and Hasan (1985), is language that is functional; that is, text is “any instance of living language that is playing some part in a context of situation. . It may be either spoken or written, or indeed in any other medium of expression that we like to think of” (p. IO). The texts analyzed in the following section include transcripts of spoken classroom discussions and written textbook materials. A basic assumption of a social semiotic perspective is that words by themselves neither have, send, or “make” meanings, nor do texts. The meanings ot words and texts are always dependent on other words and other texts. 1 show that the meanings of a single text produced in the mathematics classroom can depend on texts comprising a wide range of different mathematical genres. The principle of general intertextuality refers then to the ways in which texts make sense in relation to other texts. Threadgold ( 1986) pointed out that there is an increasing emphasis on the “importance of the intertextual in the production of coherence” (p. 52) notably in the work of Hasan (1985) and Thibault ( 1986, 199 I), as well as Lemke (1985). Lemke stated, “every text, the discourse of every occasion, makes sense in part through implicit and explicit relationships of particular kinds to other texts, to the discourse of other occasions” (p. 275). General intertextuality thus requires that meaning is studied through and across texts. It proposes that thematic formations can be represented in many texts. in different kinds of language. Why, then, do certain texts contextualize others‘? How do contextualizing practices work’? According to Lemke (I 985), the notion of general intertextuality addresses these questions, asking “which texts go together, and how:'": That

texts “go together”

means that there is some scnsc in which

contexts for each other’s

interpretation,

made by the community

through

go together is to specify meanings.

The

kinds

that socially

the interrelations

what kinds of relationships

of relationships

significant

they arc relevant

meanings

of these texts.

are being

To say how they

between them help make these

made between texts and the acts of tcxth

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related in these ways together define the system of intertextuality in the language use of a community. (p. 276) The thematic content of a lesson, that is, what is being taught, comprises patterns or systems of thematic meaning. Thematic systems are systems of meaning relations that link texts together. Lemke (1983) made a distinction between the text thematic system specific to a particular text and the intertextual thematic systems common to a set of texts. Thematic development refers to the “enactment” of these systems. The thematics of a text are its actual patterns of meaning relations. The semantic and thematic patterns and genre structures of school mathematics discourse can be understood as being built up over the production of many texts. Everything that is said is understood according to what has already been said and what is being said or presented in other texts. For example, a typical mathematics lesson on the topic of index laws, a new topic for a class, may be introduced by teacher talk, which includes worked examples written on the blackboard, small-group discussions, and then individual textbook exercises which are to be completed as homework. Each of these texts, from different genres, adds to and refines the thematic content of the lesson. Together, they construct a system of meanings about index laws. In examining some specific text-connecting practices of school mathematics, this article shows that these practices are a necessary part of the processes of constructing shared meanings and illustrates something of their complexities. TEXTUAL

ANALYSIS:

THE CASE OF FUNCTIONS

This section analyzes a variety of texts from a series of lessons on the topic of functions. An important feature of this topic is representing relations between variable quantities in which the value of one depends on another, in mathematical terms. The topic introduces students to different types of variation in situations and goes on to link these with types of functions. They learn that different rules match different functions. The topic deals with technical aspects of both symbolic and graphical representations as well as with procedures for solving equations. The texts chosen are typical of those produced in any one lesson for this particular class, described by the teacher as a “fairly able” group with some “very good” students. My concern is with how language is used to construct the thematics of a mathematics lesson; in this case, the topic is functions. How is sense made of each text? What are the thematic formations? How does a text instantiate a thematic system? What ties these different and distinct texts together‘? In what ways is the common thematic formation signaled? These questions are the focus of the following analysis. Together, this section and the next examine the intertextual relations within and across texts of various genres. The texts comprise transcripts of naturally occurring discussion and dialogue between teacher and students as well as written textual material from textbooks, student work, teacher notes, and blackboard

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work. The analysis is selective, focusing on the development of systems of thematic relations. The first stage of the analysis considers the thematics (i.e., the actual meaning relations) constructed in the language patterns of each text. It examines the textual cues to combine the thematic patterns of separate texts with the familiar thematic formations of other texts. The analysis goes on to show that the distinct patterns of meaning identified are in fact instances of a common thematic formation. The aim is to identify and illustrate features of the textconnecting practices of school mathematics. Homework Review The first text exemplifies a lesson-opening activity for the class, which usually occurs immediately following or occasionally in place of a brief oral arithmetic scsslon. Text

I 2 3 4 5 6 7

Teacher:

8

Teacher:

rule? Really?

Ryan,

would

board that WC arc dealing situation. actually

9

I

on page ninety

didn’t

don’t

numbers.

X and y table.

worry

about that,

I2

those numbers.

I3

lot faster than previous

I7

Ryan:

I8

Teacher:

Right.

now.

Ryan,

Those

who did them, ones wc’vc

OK. two.

done.

you just

I just

Then

put those littlc

comes between.

the

we’ll

Good,

generated

these

you agree

eight.

four, OK.

to the to find what‘s

What

with

running

up a

did you do?

Just doing

kept working

1 can see

n’s in so

it down

the diffcrencc

and then that was to thc--

it one more time so those people

20

get somewhere.

looking

where

All right’!

you all find this’? Did you run through

who really

believe

this just one time.

for the common

difference

it. did

hopin?

to

and all that comes

up is-

21 22

Ryan:

23

Teacher:

I

kept going down

Keep working actual

24 25

bccausc

just go straight

at a pattern of dots which

with great care.

I9

26

rcmembcr

it. It was

I tried to put in. umm. Well, pattern

I6

numbers,

like to find

For those who are struggling

numbers.

Ryan:

who got a up on the

finish-

Actually.

going on, WC are looking

Teacher:

First line

The

You might

six. Get your books out please,

II

IS

with’!

anyone

the numbcrs

a lot to do.

IO

I4

you mind putting

it’s in the textbook.

bc having Ryan:

1

Those who did it (referring tO homework). is there

Ryan:

No.

till

it down.

dots that mcan the diffcrcncc

pattern

is in

fact two?

because

because

it’s not constant.

there‘s

You can’t

tell if it’s constant

or not

only one number

Text 1 is an excerpt from a discussion between the teacher and a student. Ryan, about the previous night’s homework. The homework involved completing a number of tables contained in the course textbook De~~rloprrwntd Muthemutics 4.4 from a series designed for the Western Australian Unit Curriculum in Mathcmatics. In Text 1. Ryan is explaining to the teacher and the rest of the class how

Intertextuality

he completed the following the blackboard:

247

in School Mathematics

table, which has been copied from the textbook onto

Text

. . Shape Number

2

. . . .

. . . . 1

. . . . . . . .

. . . . 2

3

. . . . . . . . 4

5

No. of Spots 1st difference 2nd difference 1st ratio

I

The mathematical themes of Text 1 are most explicitly signaled by the use of the following technical terms: numbers, rule, x and y table, pattern of dots, di$erence pattern, common deference, and constunt. Throughout the discussion between Ryan and the teacher, particular patterns of relations between these linguistic items are constructed. These patterns, or thematic relations, are repeated and reinforced in the dialogue, although they are not always described in the same language. The first of the thematic relations in Text 1 is presented in the teacher’s opening statements in Lines 1 through 6. The language used by the teacher to structure the homework review activity also sets up a semantic relation between numbers and rule. These two terms are thematic items that form the first part of a system of thematic patterns. This particular pattern is the connection between numbers and rule. The semantic relation is that numbers lead to rules. The homework exercise provides a special instance of this relation. In this case, the numbers are not just any numbers, they are those that “we are dealing with” (Line 3). They are the “first line numbers” and they represent a particular “situation.” The first part of the thematic relations for Text 1 can be shown in Figure 1. The teacher’s response to Ryan in Lines 8 to 11 presents two other relations. The first relation is between “the numbers” and the “x and y table.” The second relation is between “a pattern of dots” and “these numbers.” Here, the teacher refers to the diagram and the table presented in the textbook (see Text 2). This other text is needed to make sense of what the teacher is saying. His instruction, “just go straight to the numbers. X and y table” (Lines 8-9), is summarizing previous work done by the students on completing such tables. He is referring to a semantic relation already constructed. The term “first line numbers” was used earlier (Lines 3-4), also signaling the relation between numbers and table. First

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248

Figure

1. Pattern of thematic relations for Text 1 (Part 1).

line numbers are those in the row labeled “No. of Spots.” The teacher’s explanation that “we are looking at a pattern of dots which generated these numbers” (Lines IO-1 1) is the only spoken reference to the textbook diagram. However, all of the discussion about the first row of numbers in the table (Lines I-14) is based on an understanding of that diagram. These two relations, between the numbers and the table and between the pattern of dots and the numbers. can be added to the thematic pattern for Text I as shown in Figure 2. The main thematic items in the remaining part of the discussion are the difference pattern and the common or constant difference. The teacher uses the term common difference, whereas Ryan uses constant, which is the term used in the textbook. The teacher and Ryan talk about the procedure for finding the rule from the numbers. Each row of numbers in the table generates the next and both Ryan (Line 17) and the teacher (Line 23) describe the procedure as “working it down.” This involves completing the x and 2’ table, filling in the difference pattern, and looking for the “common difference” (Line 20). The concern is now with the difference pattern that can be found in each row of numbers. rather than .just with the first line numbers. This procedure sets up a thematic relation between the two new thematic items, the difference pattern and the constant difference, which completes the thematic pattern of relations for Text I (see Figure 3). In Text I, several references are made to other texts. For example, the “it” in the teacher’s question “who did it” (Line 1) refers to the homework. The homcwork exercise is a written text presented in the course textbook: It comprises a diagram and a table. The procedure for completing the table was discussed in

x + y table ii

pattern

of dots

L

generates *

has

first line numbers

Figure 2. Pattern of thematic relations for Text 1 (Part 2)

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x + y table

Figure 3. Pattern of thematic relations for Text 1 (Part 3). class the previous day and a worked example was presented on the blackboard. Putting in “those little n’s” (Line 15) is a technique used in that example. In Line 13, the teacher mentions “previous ones we’ve done,” again referring to other worked examples. When the teacher mentions “looking for the common difference” (Line 20), some other text is needed to explain about constant first and second difference patterns. One such text is the written instruction to the homework exercise (see Text 7). Text 1 makes sense only in relation to these other texts.

Revision Text 3 I

Teacher:

2 3 4 5 6 7

Linda: Teacher: Linda: Teacher:

8 9 10 II 12 13 14

Stuart: Teacher: Stuart: Teacher:

What we did last week was to recognize that linear functions are patterns and from the pattern, Linda, what told us that we were dealing with a linear function? It was the same. What was the same?

The difference pattern. The difference pattern is constant and that leads us to finding the rules. Remember how we got the rules from the pattern. Could someone remind me, or remind us, how did we get the rule from the pattern when we were talking about linear functions? Half of the, umm, bottom line, wasn’t it, its The difference pattern was two, two, two, two. What did that tell us? One x umm something. Is he right?

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250

IS

Students:

16

Kim:

17

Tcachcr:

No. It’s two

x.

Is she right?

(several

IX

Well

I!,

it a littlc

more

It’s

it’s

she‘s just

20

Astra:

11

Tcachcr:

That’s

23

Astra:

If it’s

33

Teacher:

74

umm

in background

Thomas:

Two x

28

Teacher:

OK.

another

30

your

31

different. Astra:

Six

yes.

35

a

hand

Ashlcigh: Tcachcr:

3x

In fact

I

Four

x plus. is that

Students:

Yes.

Tcachcr:

OK.

Text 3

is an extract

one.

student

functions

it look?

Pretty

does it look

looh

(Several

easy, give

like

voices

mc an

like’!

Hand

up plcasc.

Someone

as you

can make

it. Oh

Maybe

answers

you

can’t

make

tell

no.

me

keep

it very

in the background.)

by t\vo. by two.

write

Y equals

three

I

what’!

Well

34

x add two.

different

could

OK.

add x. is four

43

doa

(Writing

on blackboard)

that as six on two

x. and

not that ditTcrcnt,

three Y. Another

I

As it happens.

could

is it? But

one? Another

write

that

it‘s a bit one?

x add x.

Y equals

Teacher:

how

at same time.) What

three Y. 50 it’s really

40 Ashlcish:

But all of our linear

is as different

y divided

down

43

answers

bits’!

to aomcthing.

the rule,

function.

six x divided

39

51

A

up.

x equals

Three

the other

it. OK. write

Two

which

(Several

different.

36 3-l

enc.

one.

Y cquals

34

about

add two.

Thcrc’s

20

Teachcr:

giving

of B linear

yes and no.)

she thought

x.

it’s added

to find

saying

so presumably

was it’!

about

I was to

If

example

27

four

that

around

26

33

What

not exactly

7s

31

closely.

and what

something.

in the background

her journal

by

right.

We play

students

written

Thcy’rc

x add x. But wouldn’t

write

of course three

to write

x add x. I’d

I

really

write,

no.

that

write three

x

x.

I don’t know still

linear?

all going

from a routine

to end up that kind

revision

or review

of \hapc.

session

for the class.

The

teacher is questioning students on work covered in the two previous lessons on the topic of linear functions. This session takes place I week after the lesson Text 1. It is a much longer episode of classroom discussion than Text I share a common thematic formation about functions. That is. their patterns of thematic relations are the same, even though they may be expressed in diRerent words. Text 3 reconstructs and adds to the relations presented in Text I. In Lines I through 3, the teacher summarizes “what we did last week” in the statement “linear functions are patterns. ‘. This adds a new thematic item to the involving Both

texts

thematic pattern already established for Text I This new semantic relation is elaborated further by the teacher; something about the pattern tells us whether a function is a linear function. In Text I, the word pattern was used in two ways: It described the “pattern of dots” that generated numbers and also referred to the “difference pattern.” In this case. it refers to the difference pattern. The key

Intertextualitg

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251

semantic relation being reinforced by the teacher is between linear functions and the difference pattern. This is emphasized in the brief dialogue with Linda. Linda’s response to the teacher that “it was the same” (Line 4) suggests that she shares the teacher’s thematic formation about linear functions. They are both talking about the difference pattern, although that particular term is not used until Line 6. The teacher questions Linda, getting her to elaborate her answer. “It” means the difference pattern. “The same” means constant. In Line 7, the teacher restates Linda’s initial answer in more appropriate language: “the difference pattern is constant.” This statement effectively reconstructs the relation between the difference pattern and the common difference that was established in Text 1. The same thematic relation has been presented in different language. Moreover, in Text 1, the relation was shown to exist for a certain case: It concerned the difference pattern for a particular example or situation. In Text 3, the relation is about difference patterns in general. Now, however, as the teacher reminds the students, they are “talking about linear functions.” A constant difference pattern now has two implications: The first implication is that it leads to finding the rules (Lines 7-8) and the second is that it indicates a linear function. The discussion in Lines 12 through 20 is about a specific example. Its meaning relies on two other texts: the example itself and the students’ journals in which they wrote about how they solved the problem. The issue here is not really the answer to a specific problem, but the form of a linear function. The point being made by the teacher is that “all of our linear functions look like something” (Lines 23-24). The link between linear functions and rules is made more explicitly. The teacher uses the terms almost interchangeably: Linear functions look like something and rules look like something. It seems that, for the teacher, the separate thematic systems are becoming integrated in this text, although there is no evidence that any of the students share this understanding. The examples of linear functions provided by the students in the remainder of the text are special cases of the idea that linear functions all have the same form; they all “look like something.” The teacher writes each of the different examples on the blackboard, explaining and showing how they are similar, and how they can be made to look the same. In this way, he is negotiating a common thematic system with the students. His final statement assumes that his thematics are shared by all: “They’re all going to end up that kind of shape” (Line 44). This new pattern of thematic relations in Text 3 can be described in Figure 4.

New Work Text 4

I

Teacher:

2 3 4 5

Astra:

Now all of those linear functions are often represented in a very special way by this, and I’d like you to write this down in a minute. Y = AX + B. Now there’s actually four letters in there, but different letters are doing different things. What are the A and the B doing’? Y equals what‘?

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252

Figure 4. Pattern of thematic relations for Text 3

6

Teacher:

7

x IO

Teacher: Arthur: Teacher:

13 I4 15

Linda: Kim: Teacher:

I7

Arthur: Teacher:

I9

Stuart:

20

Teacher:

Y = AX

+ B. What

OK,

arc the, what’s

the A

kind of like the general

function.

Sir.

Arthur,

what do you think’?

They

are numbers.

Ych,

that’s right.

They

are representing

numbers.

What

number

A be?

Four. TWO.

Two or three or anything? Five

million

six hundred

Is he right? Five No,

million

and seventy

four.

six hundred

and whatever?

it’s only a one digit number.

Is it, is it a linear so it could

21

function’?

be. What

about,

Y equals one hundred what’s

x plus one.

the kind of number

Yes,

you haven’t

told mc about yet? You told me a big number.

22 23

Numbers,

could

I6 I8

up there.

name for a linear Arthur:

12

it’s written

and the B, what does it stand for’? That’s

9 II

Well

Ian:

24

Teacher:

25

Students:

26

Teacher:

27

Students:

28

Teacher:

Do it in negative. Do it in negative? Yes. Is this a linear

Is it’? Yes it is. OK. numbers

29 30

Matthew:

31

Teacher:

32

Students:

33

Teacher:

35

Students:

36

Ashleigh:

37

Tcachcr:

38

Ashleigh:

could

So it could be negative.

it be that you haven‘t

What

other type\

ot

told mc about’!

Fractions. Fractions.

Could

it bc a fraction’!

You sure’! Could

this bc a linear

Yes. we’ve

34

function?

Yes!

been working

function?

Does that look like what

with‘!

No. You just put zero point five and Oh well, YCS.

if you like.

You want to change it to a decimal.

Is that OK?

Intertextuality

39 40 41 42 43

Teacher: Astra: Teacher: Ashleigh:

in School Mathematics

253

Was it OK when it was a half? No, it was harder to work with. Is that a linear function’? (Students answering yes and no in background.) That’s no. What do you think, Ashleigh, no or yes‘? That would be right.

Texts 3 and 4 are from the same lesson.

The discussion in Text 4 takes place shortly after that in Text 3. Text 4 is presented here as a separate episode because it presents a new topic to the class. The thematic content in this episode is the equation Y = AX + B. This is written on the blackboard and spoken by the teacher (Lines 2-3). This introduces a new aspect to the thematic formation already established for the topic of functions. This form of representation has not been used before. Its meaning, however, is contingent on the previous work on functions, on the thematic formation for functions presented in previous lessons, and in other texts. The link between this episode and the previous one is signaled by the teacher when he refers to “all of those linear functions.” This could mean those examples of linear functions provided by the students in Text 3. It could also mean all linear functions. Each of these possible meanings fits the system of thematic relations already constructed. Throughout this discussion, the teacher is working toward establishing the idea that Y = AX + B does represent all linear functions. He instantiates this by getting students to say what the letter A could stand for. It becomes a sort of game, with one student suggesting “five million six hundred and seventy four” (Line 17), another saying “do it in negative” (Line 23), and a third saying “fractions” (Line 30). The teacher encourages this kind of response, leading the students to conclude for themselves that A could stand for any of these kinds of numbers. “A real text will have many thematic strands, many thematic formations which it links together to make its arguments” (Lemke, 1990, p. 205). This is clearly so in Text 4. Each of the different kinds of numbers provided by the students is a thematic item that has its own thematic formation. In this text, however, they combine to form a particular set of relations, among each other, between linear functions and numbers, and between the equation Y = AX + B and what it represents. Although there are in Text 4 significant terms that are new to the students, the thematic relations they describe are the same as those in previous lessons. The Y = AX + B is now “the general name for a linear function” (Lines 7-8). It is a kind of summary statement of several of the thematic relations in other texts. A pattern of dots, numbers in the x and 4’ table, and difference pattern can all be described by the linear function Y = AX + B. So, the thematic system for Text 4, shown in Figure 5, can be seen to fit the system already established for Texts 1 and 3.

254

A. Chapman

Figure

5.

Pattern of thematic relations for Text 4

Group Work Text 5 I

2 3 4 5 6 7 x 9 10 II 12 13 I4 IS I6

Stuart:

SO you

find where

to USC two points,

it intersects. don’t

Brian:

So you need two

A’s.

Stuart:

Yeh,

one way.

well there’s

one and across one, Brian:

That’s

when

That’s

when y equals

y equals

Brian:

Ych, that’s And another

Brian:

Or you don’t

Brian:

Three.

Then

You find out where

three is then you go up

x.

three.

what we’ve

got to work

out.

way is the way Erin did it, when

you got five points.

need to use the point on the y axis.

three other points. Stuart:

it’s three and you only need

up one and across one, up one and across enc.

Stuart:

Stuart:

We know

you’?

You could just plot

so-

you work

out x equals four,

Or you could just ignore

five.

so-

the y axis.

Stuart:

Do you have to write

Brian:

The first one, and what was the other one you said‘?

Stuart:

17

Plot three other points, Then

I8

Brian:

IV

Stuart:

without

it should cross through

Are you

going to write

the ways you do?

using the axis? without

joining

them up.

the same axis.

it down’?

Ycp.

In this discussion, Teacher:

down

Work

Brian and Stuart

out in your

group

are responding

two

diRerent

ways.

to the teacher’s completely

instruction:

different

ways,

of

drawing

the graph of y equals x add three. Discuss it and then do it. but only do

it when

you all agree on it.

I

want two different

ways.

The “two different ways” are really ways that have already been demonstrated in previous lessons. The discussion between Brian and Stuart shows that they are aware of this. Each proposes a method that has been taught as part of the topic 01

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functions. There is, however, evidence in this text that they do not share a common thematic formation. They appear to be communicating successfully; they do not explicitly argue, disagree, or challenge each other’s ideas. Nevertheless, these students are constructing quite different thematic relations. Stuart begins the discussion with a reference to the rule Y = AX + B that was introduced in the previous lesson. He recognizes that Y = X + 3 is a special case of this general rule and assumes that Brian does, too. He states, “where it intersects. We know it’s three” (Line I). The graph has not yet been drawn and SO it seems likely that Stuart knows from the rule that the y intercept is three. Brian is also referring to the rule when he replies, “so you need two A’s” (Line 3). He also recognizes that Y = X + 3 is of the form Y = AX + B, but he has constructed a different meaning of what the A represents. He does not clarify his statement, nor does Stuart ask why you would need two A’s. He simply ignores Brian’s comment and goes on to outline one method of drawing the graph. Stuart shares the teacher’s thematics, presented in earlier lessons, that A represents the slope and B the y intercept of a linear graph. His method, explained in Lines 4 and 5, is to start from the point in which y = three (“you find out where three is”) and use the slope to draw the graph (“you go up one and across one, up one”). He knows that the line has a slope of 1 and knows what the line will look like. He is using what the teacher has called the “rise over run” method of determining the slope of a line. He is thus drawing together several thematic strands: about the general rule for linear functions, about the y intercept, and about the slope of a line. Brian’s reply in Line 6 shows that he does share Stuart’s thematic formation for a slope of 1. He also knows the rise over run method. He knows that the equation Y = X has a slope of 1. The point of conflict between the two students is the representation of Y = AX + B. Brian apparently sees this as general rule but he does not see a connection between the equation Y = X and the one they are attempting to draw. He does not see that Y = X and Y = X + 3 have the same slope. The students’ disagreement is evident in Lines 6 and 7. Brian objects to Stuart’s method, saying “that’s when Y equals X.” Stuart responds, “that’s when Y equals three.” Neither explains why they disagree. The argument is not resolved and neither understands the other’s point. Brian ends this part of the dialogue (“yeh, that’s what we’ve got to work out”), leaving the conflict unresolved. Stuart considers that they have already found one way of drawing the graph. He states, “and another way” (Line 9). Brian does not challenge this implication. He does not suggest a method and clearly does not follow Stuart’s method. However, he seems satisfied to move on to the next stage of the activity. The students begin to communicate more successfully at this point. Stuart suggests finding five points. This is “the way Erin did it.” Brian now refers to Stuart’s earlier method. He says quite confidently, “or you don’t need to use the point on the Y axis” (Line 10). It is not clear whether he now follows Stuart’s method, but at least part of their thematics are apparently the same.

256

A. Chapman

It is Brian who puts forward the second method. He suggests, “you could just plot three other points” (Lines IO-1 1). Stuart interrupts, showing how this method could work:“then you work out” (Line 12). Brian repeats that they do not need to use the Y axis for this method. The students are now negotiating a common thematics. Although Brian does not seem to understand the first method cxplained by Stuart, he is satisfied to accept that it is a “sufficient” answer. He also states that the second method is Stuart’s (“and what was that other one you said?“), even though it was Brian who suggested plotting three points. Stuart summarizes this second method, introducing a further aspect. “then it should cross through the same axis” (Line 17). This has not been mentioned before. Brian does not take up this point of the discussion. His concern is to write down their answers. The point of conflict in the two thematic systems operating in Text 5 is not immediately obvious because the two students do not attempt to engage with each other’s meanings. Stuart’s thematics are the same as those in the other earlier texts considered. Brian’s thematics for the first method discussed arc clearly different from Stuart’s, In the second part of the discussion. however. Brian and Stuart work toward negotiating a common thematic system. Textbook Material The previous section showed that a variety of spoken texts share a common semantic pattern of relations for the topic of functions. These patterns can be represented as a thematic formation for functions. It has been shown. too, that the meaning of spoken texts relies on other texts and that these may include written forms. In the texts of teacher and student discussion considered prcviously, references are made to written texts, including blackboard notes. homcwork, student journals, and textbook exercises. This section will show that the thematic relations identified in the spoken texts arc also reconstructed in written textbook material. It is argued that learning about functions involves identifying. constructing, and linking its thematic patterns across a range of both spoken and written texts; that is, it requires text-connecting strategies. School mathematics textbooks are notoriously difficult to read. Mouslcy and Marks (199l), for example, argued that the language of textbooks often discnpowers the student. They stated: the oral use of mathematical the language is encountered genres of explanation text.

written

in mathematical

mathematics

It is essential

mathematics.

before

text. Students also need to be familiar

with

present arguments in steps; they need to understand the

the way mathematicians decoding

language by the student needs to bc cxtensivc in (written)

then.

textbooks

and need to develop

skills

in

before they can benefit from the content of (written)

that mathematics

teacher5 train

their students

to IPA

(pp. 7.5-76)

&rot (1990) produced a linguistic analysis of the language patterns found in mathematics textbooks. She stated that the generic structure of textbook mathe-

lntertextuality

matics

includes

five phases:

257

in School Mathematics

(a) introduction,

(b) activity,

(c) information

giving,

in the instruction phase the emphasis is typically on procedure. The underlying mathematical principles are often implicit and not articulated in the wording of the written text. She surmised: (d) instruction,

and (e) practice.

According

to Gerot,

especially

Mathematicians, including those who write school textbooks, have the gift of “understanding visually” and find it difficult to appreciate what it is like -not to “see” in this way. But for those of us without this gift. more explicit verbalisation of principles may be necessary for mathematics to “make sense.” (p. 7) School mathematics symbolic, pictorial,

textbooks typically contain and graphical representations.

a mix of verbal language and Written mathematical mate-

quite dense and thus demands a more exacting style of reading than other types of printed material (Chapman, 1988). I argued elsewhere (Chapman & Lee, 1990) that mathematical ideas and representations need to be read very differently from other kinds of linguistic items. The textual signals to construct meanings in mathematics textbook material are likely to be different from those in spoken texts. According to Lemke (1989), the problem of learning through written texts is “fundamentally a problem of translating the patterns of written language into those of spoken language” (p. 136). He stated that teachers and students can make text “talk” in two ways: They can read a written text out loud so that it becomes a part of classroom dialogue or they can elaborate and comment on it in classroom discussions, integrating the formal written language of the subject into their natural ways of talking. Sense is made of written text by translating its meanings into the “more comfortable” patterns of spoken language. Text 6 is from the same lesson and concerns the same homework exercise as Text 1. It becomes evident that the teacher and students are building connections between the words of the textbook and more informal language. Also, they are working toward speaking the thematic pattern of the written textbook material presented in Texts 7 and 8 to follow. rial is usually

1

2 3 4 5 6 7 8 9 IO I1

Teacher: Arthur: Teacher: Stuart: Teacher: Stuart: Teacher:

Text 6 What were the distinguishing features of our linear functions? Arthur? A straight line. They formed a straight line. What else can you tell me about linear functions in terms of the tables of values? It always went up by the same. It always went up by the same. So how did you think that, how did you know that it always went up by the same amount? Because that’s what the rule says. How in the table did you see that it always went up by the same? Go on. I’m just looking for the key word here. Helse?

258

12 13 14 I5 I6

A. Chapman

Helsc: Teacher: Stuart: Tcachcr:

Constant. First the pattern was constant. OK. What was the second one we looked at? Tell us about the quadratic function. The second line is the one that’s the ~nc. OK. So two lines to get to the constant pattern.

The teacher asks Arthur to name the “distinguishing features” of linear functions (Lines l-2). Arthur gives one feature but this is not what the teacher wants. Arthur’s answer is correct, though. This is signaled by the teacher who repeats Arthur’s response but restates it as a complete sentence. This is a fact; it is something that should be remembered. However, the teacher wants the students to provide some other information. He points out that he is concerned with the relation between linear functions and tables of values (Lines 4-S). Stuart’s answer that “it always went up by the same” (Line 6) makes sense in the context of the preceding discussion of Text 1 about difference patterns. However, the teacher asks Stuart to elaborate on this, asking. “How do you know that’?” He is apparently seeking a more appropriate form of expression. In fact, he makes this quite explicit in Line I I : “I’m just looking for the key word here.” Stuart resists this. avoiding more formal mathematical words. Helse provides the “key word” sought by the teacher, ‘* constant” (Line 12). This leads the teacher to state the thematic relation between the difference pattern and types of functions. Stuart apparently understands this relation. His thcmatics seem to be the same as the tcachcr’s. although he uses different language to explain the relation bctwccn the difference pattern and quadratic functions. Stuart’s statement, “The second line is the one that’s the same” (Line 15) and the teacher’s, %o two lines to get to the constant pattern” (Line 16) are different expressions of the same thematic pattern, presented in the textbook in the following way: Text 7 For each of the following which

accompanies

linear,

quadratic

Remember:

it. Determine

linear types

exponential try to find

the table

the situation

is of a

type.

have types

copy and complete whether

or exponential

quadratic

Then

patterns,

a constant have

types

first difference,

a constant

have

a constant

second

difference,

first ratio.

a rule for each situation.

Text 7 is very clearly a mathematical text. This is evidenced by a number of features. First, it contains some highly technical terms. It has been shown that these mathematical terms do become part of natural classroom discussion but

Intertextuality

259

in School Mathematics

usually after much more informal talking around the topic. In Text 3, for example, a difference pattern was just a “pattern” and a constant first difference was described as being “the same.” The semantic relation in that text and Text 7 is the same but expressed in different kinds of language, which can be understood as more or less technical. A second feature that marks Text 7 as mathematical is the use of definitions. Martin (1989) stated that “definitions are a special type of relational clause. [They relate] previously defined technical terms to new technical ones” (p. 39). The three definitions of types of functions in Text 7 do just that. Text 8 shows how the situation types are defined in the preceding exercise of the textbook. The technical terms constantfirst d@rence, constmt setntzd di$erence, and canstunt ,first ratio are introduced in this exercise. They refer to the rows of the tables to be completed by students. The rows are labeled ‘“1st difference,” ““2nd difference, ” and “1st ratio.” Thus, the tables relate these new terms to each other and the definitions relate new terms to old. Text 8 Let us investigate Mathematical (a) LINEAR

what

type of relationship

Development relationships

units, have

this is. From

we know

a constant

previous

that:

first difference

first difference gives us the SLOPE or GRADIENT

(and that

of the linear

graph). (b) QUADRATIC

relationships

have

a constant

second

difference.

Martin (1989) also pointed out that “in order to understand technical discourse both the definitions and the relationships among what is defined are critical” (p. 39). In terms of constructing a thematic formation, the relations contained in the definitions are highly complex and rely heavily on other relations and other texts. To construct a meaning for constant first difference, students first have to complete, or at least read. some of the tables in the exercise. They need to have constructed meanings for the terms difference patterns. constant differences, and first and second differences, as well as relations between them. The patterns of relations involved in Text 6 rely heavily on those in Texts 7 and 8 and can be shown in Figure 6. A THEMATIC

FORMATION

FOR FUNCTIONS

Each of the texts previously considered constructs a pattern of meaning relations for the topic of functions. The thematic analysis has shown that these patterns are in fact different representations of at least parts of the same thematic formation. The common semantic patterns that have been identified can now be represented as a composite thematic formation for functions as shown in Figure 7.

A. Chapman

260

rule

difference pattern

I

Figure 6. Pattern of thematic relations for Text 6.

The study of spoken texts has illustrated ways in which the teacher and students have constructed the patterns of meaning relations for functions. Each text uses language in specific ways and students need to pick up on these textual cues to construct the thematics of the topic. The written texts considered are all from the same course textbook and are all contextualized by the spoken texts. The analysis has illustrated that the patterns of thematic relations identified in the spoken texts are reconstructed in different forms in the written material. Lemkc’s (1989) notion of “making text talk” (p. 136) has been used as a basis for showing that the teacher and students connect the thematics of spoken and written texts for one school mathematics topic and how this is carried out. For example, the last

I pattern of dots 1-q 1

I

x + y table

I

I

I

I I rows

I

of numbers

I ’

I /-

( t

/-

t

t

rule

difference pattern

piiEz--quadraiic-_------_.qD] Ft-------Figure 7. Thematic formation for functions

I

I I I ,

Intertextuality

in School Mathematics

261

two texts (both comprising technical language) exemplify the kinds of language found in mathematics textbooks. It has been shown that these formal, definitional texts rely heavily on other meaning relations and other texts in the same ways as the spoken texts. They repeat the common semantic patterns of the topic of functions, providing a more formal or mathematical use of language to represent the thematic formation. The language strategies used to construct the thematics of each text are complex. The meanings and meaning relations do not unfold sequentially. Rather. they overlap, are repeated, and reinforce to build up an overall framework. The main focus has been on the thematic patterns of texts and it is clear that these patterns involve genre structures and semantic strategies. To make sense of any particular text, the student needs to connect it to other texts, often from other genres. Lemke (1982) pointed out that a student who has mastered a semantic pattern usually has no trouble matching it to new words in different texts. He stated, too, that misunderstandings occur when speakers approach a topic with different semantic patterns. This is so in the texts considered here.

CONCLUSION This article has demonstrated the operation of text-connecting practices in the language of school mathematics. It has shown that a common thematic formation for a mathematics topic can be repeated in different ways in different kinds of language, both spoken and written, which can be described as more mathematical or less mathematical. The analytic section has identified and illustrated some specific textconnecting practices used to construct a thematic formation for the topic of functions. A close analysis of the thematics of a range of texts of different genres has shown that they all repeat the common semantic patterns of the same topic. It has shown, too, that different texts use different linguistic resources to construct common thematic relations. It seems that certain kinds of linguistic information are likely to be helpful to students as they learn intertextually in mathematics. Connections with other texts and other contexts are essentially what build the thematic patterns of individual texts into a thematic formation. These connections can be made through explicit reference to familiar thematic patterns and relations. Thematic items can become naturalized into the everyday language of the classroom through regular use in different forms. The transformation from a nonmathematical expression into what is clearly a more mathematical one is a popular and powerful strategy for reconstructing a thematic relation. Highlighting key topic-specific words, and thus making connections between spoken and written texts, is a further way to connect texts and develop thematics. If students are to make sense of the texts they encounter and if they are to construct the shared meanings of school mathematics, then they need the appro-

262

A. Chapman

priate text-connecting strategies. These strategies include a knowledge and understanding of the different genre structures of mathematics. Learners of mathematics also need strategies to construct the semantic relations of a thematic formation, for example, knowledge of the thematic items of mathematics and the ability to construct relations between them.

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