Université Laurentienne

Ontario, Canada

* Abstract: Taking
generalization as a cultural semiotic problem, that is, a problem
about meaning co-construction occurring in the overlapping
territories of writing and speech, this paper attempts to study
generalization as a mathematical action unfolding in a classroom
discursive Bakhtinian ‘text’ jointly written by teachers
and students in the course of mediated activities. In the case that
we shall consider here, what is at stake is the construction and the
meaning, in a grade 8 classroom, of a new mathematical object
—that of the general term of a sequence or pattern. We shall
focus on the problem of how generalization finds expression in
processes of sign use (particularly *sign understanding

This paper is part of an ongoing
research program dealing with the students’ processes of
symbolizing in algebra^{1}. By students’ processes of
symbolizing we mean the different ways in which students come to
*understand, use* and *produce* signs. Our work is embedded
in a post-Vygotskian semiotic theoretical framework that we
elaborated elsewhere (Radford 1998, in print) in which signs are seen
as psychological tools, symbolically loaded and intimately linked to
the actions that the individuals carry out in their activities.
Within this theoretical context, ways of symbolizing (Nemirovsky
1994) are not considered as acultural**,** pre-given processes.
Instead, we consider them as instances of the general modes of
signifying resulting of the juncture of sign-mediated activities of
the individuals and the *Cultural Semiotic System* (e.g.
beliefs, patterns of meaning-making; see Radford 1998) in which
activities are subsumed. Mathematical generalizations as well as
other mathematical activities are framed by specific and culturally
accepted ways of symbolizing. This point can be made clearer if we
consider sign use in the historical example of the study of numbers
in Antiquity. While mathematicians in the Pythagorean tradition,
legitimately used pebbles to investigate some properties of numbers,
in Euclid’s *Elements* not only the actual pebbles but also
any iconic representation of them was completely dismissed and
replaced by a referential, non-operational sign-segments/letters
language couched in a deductive line of reasoning. The deductive
Greek mathematical style was linked, as A. Szabó suggested, to
the Eleatan distinction between true knowledge and appearance and the
consequent rejection of the sensible world as carrier of knowledge.
Moreover, and most important for our discussion, the Eleatan beliefs
legitimized new ways of symbolizing which authorized certain rules of
sign use and excluded others. How the Euclidean "mathematical
generality" could be expressed depended on the Greek conception of
the concrete and the abstract, the historical availability of the
Phoenician letter-based alphabet adopted by the Greeks (a
letter-based written language which was completely different from,
for instance, the syllabic Akkadian cuneiform language of the
Babylonian scribes) and on the accepted cultural normative dimension
in which the use of signs was caught. Let us consider a short
example. Proposition 21, Book IX of Euclid's *Elements* reads as
follows:

If as many even numbers as we please be added together, the whole is even.

For let as many even numbers as we please, AB, BC, CD, DE, be added together; I say that the whole AE is even. For, since each of the numbers AB, BC, CD, DE is even, it has a half part; [VII. Def. 6] so that the whole AE also has a half part.

But an even number is that which is divisible into two equal parts [id.]; therefore AE is even (Heath 1956, Vol. II, p. 413).

Euclid expresses here "generality" in
natural language as a volitive potential action rendered by the
comparative formula "as many even numbers as we please". And, within
the Euclidean semiotics, the letters allow combinations (in fact
assemblages, e.g. AB) which denote segments that stand for
non-particular numbers. Interestingly enough, the proof was not
recognized (either by Euclid or his later commentators) as lacking
generality despite** **the fact that a drawn segment unavoidably
has a particular length as well as the fact that it was actually
based on only *four* numbers. As far as we know, the proof was
considered completely general by the canons of Greek mathematical
thought. As in contemporary classrooms, modes of symbolizing and
expressing generality in Antiquity were shaped by the master’s
and students’ beliefs about mathematics, and their mutual
understanding and acceptance of legitimizing procedures about
mathematical symbolization.

In this paper we shall deal with a
problem which arises in the algebraic study of patterns, namely, that
of generalization. Ordinarily, in such cases, because of curricular
requirements (as is the case in the current Ontario curriculum for
Junior High-School), generalization is expressed through the
semiotics of the algebraic language. Of course, a great deal of
experimental research has shown that the algebraic expression of
generalization is very difficult for students who are still acquiring
the mastering of the algebraic language (see e.g. Rico *et al*.
1996). In accordance with our theoretical framework (Radford 1998, in
print), we will attempt to explore generalization as a semiotic
problem, that is, a problem about meaning co-construction by teachers
and students in the course of mediated activities. We shall focus on
the construction and the meaning, in a grade 8 classroom, of a new
mathematical object—that of the general term of a sequence or
pattern. We are particularly interested in the problem of how
generalization finds expression in processes of sign use. Since the
general term cannot be ostensively pointed to as one can point to a
door or to a desk, the semiotic construction of such a mathematical
object acquires a particular didactic interest.

In our research program we are
accompanying for three years some 120 students and 6 teachers in the
teaching and learning of algebra. This task includes the
teachers’, researcher’s and assistants’ joint
elaboration of general and particular goals, the joint elaboration of
teaching and learning settings, the video-taping of the lessons,
discussions, and feedback. The teaching settings have been elaborated
in such a way that the students (who are presently in Grade 8) work
together in small groups; then the teacher conducts a general
discussion allowing the students to expose, confront and discuss
their different achieved solutions. In general terms, we are
interested in investigating the students’ processes of
symbolizing in specific teaching settings about patterns on the one
hand, and equations and inequations on the other. In this paper,
however, we shall focus solely on the students’ and
teacher’s co-constructive semiotic expressions of the "general
term" of a sequence or pattern. The results that we shall present
here come from an interpretative, descriptive protocol analysis
(Fairclough 1995, Moerman 1988). Because of the length requirements
of the article, we shall limit ourselves to the protocol analysis of
one of our student groups. The protocol analysis will attempt to
disentangle texture forms underlying the process of sign use
(particularly *sign* *understanding* and *sign
production*) in terms of the conveyed meaning and of the classroom
use of utterances genres (e.g. reading, confronting, requesting,
informing). Our question can then** **be explicitly formulated in
the following terms: how do students’ and teachers’ voices
and writings find their way in the construction of the new object
(from the student’s perspective) of the general term of a
pattern or sequence? How do teacher and students deal with the
concrete and the abstract in pattern problems? In its most general
terms, and taking the term ‘rhetoric’ as a mode of
discourse or text making, the question is: How does the rhetoric of
generalization take place in the classroom?

The students were asked to work in
groups to solve some problems about patterns. In previous activities
they investigated some patterns and had to provide answers to
questions like *a* and *b* shown below. Questions *c*
and *d* required a new kind of symbolic understanding. For the
sake of brevity, we will consider here only some excerpts of the
episode concerning one 3-student group discussion of questions
*c* and *d*. Let us nevertheless mention that, although
questions *a* and *b* led to different understandings of
how to investigate patterns, the students did not raise problems
concerning issues on generalization. The students kept focused on
concrete issues raised by those particular questions. The case for
questions *c* and *d* was very different.

a) How many circles would you have b) How many circles would you have

* in the bottom row of figure number 6? * in the bottom row of figure number 11?

* in the top row of figure number 6? * in the top row of figure number 11?

* in total in figure number 6? * in total in figure number 11?

c) How many circles would the top row d) How many circles would figure number

of figure number "n" have? "n" have in total? Explain your answer!

Time Line dialog / remarks1:41 (21)

student 2:(he writes the answer to the third part of question b while saying)in total that comes to 24. Wow! This is easy!(Now he reads question c)How many circles would the top row of figure number … What? OK. Somebody else!1:59 (22)

student 1:(reads the question.)How many circles…[…] What does it mean?2:07 (26)

student 2: I don’t know. (hitting the sheet with his pencil)2:13 (27)

student 1: What’s figure n? /(inaudible)2:22 (28)

student 1: (talking to student 2) Shut up! I’m going to kill you. … n is what letter in the alphabet?2:33 (29)

student 2: (talking to student 1) Ask the teacher.

In this passage the students are
trying to make sense of the expression "figure number n" contained in
question *c*. As we noticed elsewhere (Radford 1996), the
general term of a geometric or arithmetic pattern cannot be
explicitly expressed within the semiotic system (SS) of the objects
of the pattern itself. Even to pose the question, it is necessary to
go "out" of the first SS (which will include, in the case of
elementary school arithmetic, the basic "well-formed" expressions
using the ten digits and certain signs like those required for the
elementary numerical operations, equality and so on) and to rely on
another richer SS (e.g. a meta-language). In the case of our text, we
had recourse to the algebraic language to talk about the general
term. As the engineering of the problem given to the students
suggests, the idea of generalization that we decide to use resides in
an experiential dynamics attempting to go beyond the concrete terms
of arithmetic. As expected, the transcript indicates, however, that
the students’ understanding of the question remained
circumscribed to their arithmetical experience. We reach here a nodal
point in the development of the classroom activity whose unraveling
will require the elaboration of new meanings. While student 2 bluntly
abandons the quest for meaning (an action accompanied by exasperation
as line 26 suggests), student 1 started a cardinal-arithmetic plan:
to display the letters of the alphabet and to figure out what
position n occupies in that order:

2:54 (32)student 1: How many circles would the top row of figure 14 have? n is fourteen.3:00 (33)

student 2: No it’s not!3:01 (34)

student 1: Yeah it is!3:02 (35)

student 3: What is n?(asking the teacher who coincidentally is walking by)3:04 (36)

student 2:(talking to the teacher)What is n? We do not know.3:08 (37)

teacher:(turning the page and reading the question aloud)How many circles would the top row of figure number n have?3:13 (38)

student 3: What is n?3:15 (39)

student 1: n is fourteen because n is the fourteenth letter of the alphabet. Right?3:20 (40)

student 2:(counting aloud the letters that student 1 wrote on the table previously) one, two three, four, five, six, seven, eight, nine, ten, eleven, twelve, thirteen, fourteen.

Student 2 does not agree with student
1’s interpretation about what n is and his arithmetical
rephrasing of the question ("confronting" utterance in line 33). The
arrival of the teacher serves as a potential way to overcome the
conflict. The teacher, nevertheless, offers as an answer a re-reading
of the question only. Interestingly (and probably because of the
teacher’s laconic** **answer) student 2 seems to change his
mind and to agree in investigating student 1’s idea; thus he
starts counting the letters of the alphabet. Noticing that the
students are taking an unexpected path that may take them away from
the intended algebraic meaning, the teacher consents to explain a bit
further:

3:28 (41)teacher: "n" is meant to be any number.3:32 (42)

student 2: OK.3:33 (43)

student 1: n is what?3:35 (44)

teacher: Any number(in the meantime student 3 comes back to question a.)3:39 (45)

student 1: I don’t understand.3:41 (46)

teacher: You don’t understand?3:42 (47)

student 1: No. […]3:44 (49)

teacher:(talking to student 3)Do you understand what n is?3:45 (50)

student 3: Which one?(pointing to the figures on the sheet)this, this or this?3:46 (51)

teacher: It does not matter which one.

The rhetoric of generalization has
now taken a different turn. The teacher launches the understanding of
"n" as "*n being any number*". The students’ reactions show
that there is a tremendous difficulty in constructing this specific
meaning. Student 3’s answer (line 50) suggests that this
difficulty is linked to a very specific semiotic problem that we will
term as the "multiple representation problem". Since sign-numbers, in
the semiotics of arithmetic, generally refer to a single object (i.e.
although in some contextual instances the number five may be
represented as e.g. 6-1, nobody will agree that, in the base 10
arithmetic, two different ‘basic’ representations like "7"
and "5" represent the same number) it appears very hard to conceive
"n" (which by the way as sign has a similar ‘basic’
iconicity as "7" or "5") as representing more than one number (see
line 50). This is corroborated by the following lines:

4:10 (53)teacher:(talking to student 2 after a long period in which the students remained silent)OK. There, do you have an idea what is n?4:12 (54)

student 1: Fourteen.4:13 (55)

teacher: It may be fourteen …4:14 (56)

student 2:(interrupting)any number?4:15 (57)

teacher:(continuing the explanation)… it may be 18, it may be 25…4:18 (58)

student 1: Oh! That can be any number?4:19 (59)

student 2:(interrupting)The number that we decide!4:20 (60)

student 1: OK then,(taking the sheet)OK, n can be … uhh…4:26 (61)

student 2: Twelve.4:27 (62)

student 1: Yeah.4:28 (63)

teacher: But … yeah. What were you going to write?4:31 (64)

student 1: 12.4:32 (65)

student 2: 12.

The teacher’s attempt failed.
The proposed meaning for n as being "any number" is interpreted as an
arbitrary but *concrete* number ("informing" utterances, lines
61, 64-65). The teacher tries to give meaning to the expression
conveying the generalization by re-investing the students’
arithmetic point of view in a way which is still coherent with the
global plan to introduce the general term in the context of the
classroom setting. The teacher’s voice hence acquires a specific
tone made up of the pedagogical plan and the students’
contextual voices. The Bakhtinian text in which generalization is
being written appears to be heterogeneous in its meaning. Realizing
that things had not turned out as expected, in the next line the
teacher launches a rescue mission from where the wanted meaning could
properly arise:

4:33 (66)teacher: And if you leave it to say any number. How can we find … how can we find the number of circles for any term of the sequence (making a sign with the hands as if going from one term to the next)4:51 (67)

student 2: Figure n? There is no figure n!4:54 (68)

student 1: (talking to student 2) He just explained it! N is whatever you want it to be.4:57 (69)

student 2:(talking when student 1 is still talking)What is it?5:01 (70)

student 1: OK. Umm … seven.(writing on the sheet)5:10 (71)

student 2: Not on top! It’s seven circles(taking the sheet and looking at the figures)5:13 (72)

student 1: Yeah! And in the bottom is 5 circles!5:21 (73)

student 2:(writes the answer and starts reading the next question)How many circles would …(inaudible)… 12 circles(writing the answer).

Speech does not unfold alone. Speech
unfolds accompanied by other semiotic systems, for instance systems
of gestures that we make with our hands and arms (see e.g.
Leroi-Gorhan 1964). When we make gestures, the hands can be used to
produce signs by e.g. sketching objects (Kendon 1993), while in
certain cases concrete objects can be used as metaphors of absent
objects (an instrumental strategy generally employed and which
becomes a cornerstone in the development of sign systems with deaf
children). Gestures form a sign system with its own syntax and
meaning that afford the production of texts. In previous activities,
we frequently saw our students pointing to a concrete figure (the
third figure of a pattern, for instance) to refer in fact to the
100^{th} figure. And in the case of the episode that we are
discussing here, the teacher makes an intensive use of gestures (line
66) to try to complement the sense of the expression "any number"
—an expression whose even most forceful utterance cannot reach
the students’ understanding yet. Indeed, the students keep the
arithmetic meanings for the relation between "n", "figure n", and
"any number". We should note at this point that student 2 is clearly
uncomfortable with their general understanding of "figure n". There
is something that does not fit the modes of meaning generation as
being used in their classroom culture. In the subjective
understanding of student 2, the order of discourse (Foucault 1971),
as legitimized by the discursive practices of the classroom cultural
institution and instantiated here by the teacher’s remarks seems
to point to a different way to interpret "n". We may say that, for
him, if n is meant to be any concrete intended number, as student 1
is proposing, then, according to the classroom culture, the teacher
could have stated this clearly instead of using such a complicated
phrasing. Student 2 is doubtful and this doubt appears as something
very important for the future of the meaning negotiation process.
Notice that the conflict between students 1 and 2 seen in lines
27-28, 32-34, arises differently here. In line 68, student 1
re-interprets the teacher’s previous explanations as confirming
his own arithmetic interpretation and challenges student 2 with an
authoritative argument ("[the teacher] just explained it!").
Seeing this, the teacher decides to intervene again:

5:42 (74)teacher: So, uh…(looking at the sheet) Wait, wait, wait! But for any number…. There you did it for seven circles, but if seven… for any …5:52 (75)

student 2: (showing the sheet with his pencil) You add 2 to the number on the bottom… subtract….oh no, you add 2 to the number on top. If it is seven, the number like this what I …(inaudible)

As the dialog suggests, the
opening towards a new understanding is not made possible through a
discussion on a concrete example but through the *prise de
conscience* of an action previously undertaken (in solving
question *a* and *b* but also in many lines of the dialog
presented here, e.g. lines 71, 72). The action is now formulated
*not* as a *concrete action* within arithmetic (which would
give as a result a concrete number, as in lines 71 and 72) but as a
*potential action* in the metacode of natural language. As we
can see, the new mathematical object is constructed with words: "You
add 2 to the number on the bottom…". What we call "generality"
is trapped here in the expression "the number on the bottom"
—an** **expression that keeps all the sensuality of the
figures in the space— and the operation of adding ("You add 2")
to which is submitted this unutterable number ("the number on the
bottom") within the elementary semiotic system of school
arithmetic.

But what is it that finally made possible the negotiation of meaning? The answer resides not in the students’ suddenly grasping the teacher’s intentions but in the teacher’s continuous (polite, encouraging but always clear) rejection of the students’ solutions and the students’ will to search for alternative understandings. The construction of the potential action with words is pushed further by the teacher in order to end up with a mathematical formula:

6:01 (76)teacher: OK. Could you put this in a formula?6:04 (77)

student 3: Uhhh …6:05 (78)

teacher: … using n.6:06 (79)

student 3: Uhh … it’s the term times two plus two.6:10 (80)

student 2: The term times two plus two?6:12 (81)

student 3: (showing the figures with his pencil)Uhhh … 2 times 6 … 2 times 3 is 6, plus two …6:21 (82)

teacher: Could you say that again, please?6:23 (83)

student 3: Yeah. The term times two plus two(student 2 writes the explanation)/ […]6:37 (86)

student 1:(reading the answer)OK. The term times two plus two.

The students finish by writing: "n2 + 2 = ".

The word "term" (which emerges as a sign representing the previous term "the number on the bottom") is not used correctly by the students from a mathematical point of view. There is a confusion between the term and its rank. Nevertheless, the attempted meaning was functionally clear. It is worth noticing that the word "term" comes first to be used by the students as a tool that allows a refinement in the construction of the object. The use of words seems to be similar to that of concrete tools in apprentices. At first the tool (in terms of the "specialist’s norms") is used awkwardly and only later can one use it with progressive mastery.

The concept of general term appeared
as a potential action bearing the concrete characteristics of actions
previously carried out in the social plane undergoing an
internalization (in the neo-Vygotsky’s sense given in Radford
1998) *through* and *by* signs (in this case words and
mathematical signs, whether iconic or arithmetic ones). Such a
potential action —which seizes the actual form of the
generalization— is the particular expression of concrete actions
as afforded by the students’ mediated activity (not only by
speech but by writing and the related cultural artefacts allowing it,
e.g. the sheet and the pencil, the latter functioning as a key
instrument in deictic gestures, as in the crucial line 75) arising in
the course of their reflections to solve the problem. The
students’ reflections and their understanding and production of
signs are embedded in discursive schemes and discourse orders
prevailing in the classroom according to its own culture.

As we have seen, the potential action
making possible the overstepping of concrete arithmetic thinking and
the reaching of generalization finds expression in the semiotics of
the concrete actions and the mode of thinking thus produced. Contrary
to the traditional idea, generalization is not something dealing with
the abstract and its evacuation of the context but a different
contextual semiotic expression of previous actions, which afford the
potential action (for instance, giving sense and virtually existence
to it). It is enlightening to remark at this point of our discussion
that Euclid’s proposition quoted in the introduction
also
bears this distinctive trait of
generalization as a potential action that, figuratively speaking,
still has the sent of the concrete Pythagorean actions from where it
emerged. Generalization is not a mere act of abstraction from the
concrete; indeed, generalization keeps a genetic connection to the
concrete according to the mediated system of individuals’
activities and the epistemic and symbolic structure of these. In
turn, as paradoxical as it may seem, the generalizing potential
action, even without being *there*, is already producing the
concrete actions. Indeed, without being explicitly there, the
potential action is already present, making possible that the sixth,
seventh or any other term be investigated in the very same form.
Beyond their synchronic temporal dimension, the concrete and the
abstract bear a dialectical relation, in which they mutually
condition each other within the limits traced by the historical and
cultural rationality of the individuals and the semiotic systems that
the individuals are continually re- and co-creating.

As we saw, the web of possibilities
from where generalization takes place is co-formed and revealed in
the texture of the text that the teacher and the students deploy in
their search for meaning. In the particular case studied here, we saw
how meaning shifted from "figure n" to "n" to "any number" to "any
arbitrary but concrete number" until they overcame the "multiple
representation problem" and reached an algebraic "public" (Ernest
1998) standard meaning. We also saw that this was done by using
different utterance genres ranging from reading ("R" e.g. line 22),
requesting ("Q" e.g. line 36), confronting ("C" e.g. line 33),
explaining ("E" e.g. line 44), acquiescing ("A" e.g. line 42),
informing ("I", e.g. line 45)^{2} . The students’
production of signs in the formula was mediated by speech and its
written form. After uttering the formula, the students wrote it in
natural language as "the term x 2 + 2" and then as "n x 2 n2+2 = " .
It is worthwhile to note that Vygotsky suggested that "
[u]nderstanding written language is done through oral speech,
but gradually this path is shortened, the intermediate link in the
form of oral speech drops away and written language becomes a direct
symbol just as understandable as oral speech." (1997, p. 142). The
fate of the students’ understanding of algebraic language seems
to be the same, that is, it will be couched in speech (and the
accompanying semiotic systems) and only later will it become a kind
of autonomous semiotic action. Indeed, signs (like "n" in the
students’ formula), we would like to insist in closing this
paper, are but the result of semiotic contractions of actions
(concrete or intellectual as outer or inner speech) previously
carried out in the social plane.

A research program funded by the Social Sciences and Humanities Research Council of Canada, grant number 410-98-1287.

The number of occurrences of types of utterances are as follows (notice, however, that a same utterance may belong to more than one category depending on its pragmatic dimension).

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