*Anna Sfard, Pearla Nesher, Leen Streefland,**Paul Cobb, John Mason*

**A. Why do we believe in learning mathematics through
conversation? An introduction**

**Anna Sfard**

**There seems to be a strong consensus among educators
about the need to foster student’s ability to ‘talk mathematics’. The Curriculum
and Evaluation Standards for School Mathematics (NCTM, 1989) include
‘learning to communicate mathematically’ among their five ‘new goals for
the student’:**

**Most of us would accept the claims about the need to
foster students’ ability to speak mathematically as a basic, almost self-evident
truth. It is the goal of the present discussion to turn the seemingly obvious
into an object of critical inspection. In what follows, the various authors
try to unpack their respective faiths in the importance of mathematical
conversation in an effort either to reinforce it or to refute it through
a disciplined analysis. At the outset, I scrutinise three basic arguments
which can often be heard from advocates of mathematical communication,
coming from well-developed, overarching conceptual frameworks which attract
many followers. In spite of such impressive foundations, on closer examination
each one of the three lines of argument reveals some weaknesses.**

*1. A cognitivist argument*

**As students communicate their ideas, explain Standards’
authors, "they learn to clarify, refine, and consolidate their thinking"
(p. 6). This clearly implies that the ability to participate in mathematical
conversation is to be promoted not only for its own sake, but also for
its expected effects on the process of learning and on the quality of the
resulting knowledge. Journals in mathematics education abound in article
titles which bring a similar message: for example, ‘On the learning of
mathematics through conversation’ (Haroutunian-Gordon and Tartakoff, 1996),
‘Reflection, communication, and learning mathematics’ (Wistedt, 1994) and
‘Journal writing and learning mathematics’ (Waywood, 1992).**

**In short, mathematical conversation is believed to
be good for your mathematical thinking. The questions which should be answered
by those concerned with cognition are of an intricate nature and require
a deep insight into the workings of human mind.**

**• What are the relations between such different cognitive
procedures as languaging and symbolising on the one hand, and conceptualising
and reasoning on the other?**

**• What does our ability to solve mathematical problems
have to do with our capacity for communicating mathematics to others?**

*2. An interactionist argument*

**For a steadily growing number of researchers, the idea
of learning through conversation is a natural by-product of the conception
of learning as an initiation into a ‘community of practice’ (Lampert, 1990;
Lave and Wenger, 1991). The essence of the argument can be presented as
follows. Mathematics students are beginning practitioners or, as Streefland
(1996a) calls them, ‘junior researchers’. If we want to have them act accordingly,
it is our job as teachers to turn the classroom into a ‘community of inquiry’
(Schoenfeld, 1996), one which would be as close as possible in its norms
and practices to those of ‘expert practitioners’. So far, so good. The
argument certainly sounds convincing in the eyes of those who adopt the
metaphor of learning as ‘legitimate peripheral participation’ in such a
community. The only remaining question is what should count as mathematical
practice.**

**As it turns out, if talking and communicating are the
hallmarks of the educators’ vision of this practice, then those ‘expert
practitioners’ are likely to take exception with the whole project. Indeed,
we require our ‘junior researchers" to communicate and verify ideas among
themselves, first in small collaborative groups, and then again, in what
has come to be known as ‘whole-class discussion’.**

**Communities of mathematicians, however, do not necessarily
work this way. On the contrary, the loneliness of the professional mathematician
is notorious. The motif of isolation and lack of meaningful communication
with others returns time and again in mathematicians’ accounts of their
own practice. Here is a representative example, in the form of advice to
a beginner, coming from a well-known ‘expert practitioner’, Camille Jordan.**

*3. A neo-pragmatist argument*

**The third argument for the centrality of conversation
derives from another new metaphor for knowledge and knowing, the same one
which gave rise to the now popular discursive approach to research on thinking
and mind (Harré and Gillet, 1995). Rather than merely treat conversation
as a secure route to knowing, more and more thinkers equate knowledge
with conversation (Rorty, 1979; Foucault, 1972; for a survey of relevant
literature, see Ernest, 1993). The gist of the idea is epitomised by Rorty
in the following statement:**

**Mathematics is certainly a conversation in this last
sense, but is it also one in the first sense? Again, mathematicians may
not be too eager to put an equality sign between their professional practices
and the activity of ‘interactive oral exchange’. This, of course, does
not mean that language and communication do not have any role in learning
mathematics. It only implies that those who try to substantiate their belief
in the power of talking with the metaphorical equation ‘knowledge = conversation’
may not be using the right type of argument.**

**Finaly, here is the question that was posed to the
panel.**

**Pearla Nesher**

**In response to this question, I would like to make
a distinction between talking mathematically and talking about
mathematics. I will start with Anna’s cognitivist argument, which is
based on the connection between language and thought: language is the major
route to the articulation of ideas.**

**The basic question which is raised within this argument
is: what ‘language’ are we talking about? Do we talk about the natural
language that serves us for natural communication in our everyday life,
in classroom conversations, in describing situations; or about the formal
language of mathematics? These two are not the same. I am worried that
speaking about ‘language’ and ‘communication’, in general terms, contributes
to the fuzziness and ambiguity of the discussion. The NCTM Standards speak
clearly about mathematics as the language of signs, symbols, etc. Symbols
and signs do not form a language unless they stand for something. They,
of course, express thought, and in our case these are thoughts about the
mathematical concepts, objects, relations, etc.**

**When problem situations are discussed in mathematics
classrooms, several levels of conversation are going on at the same time.
There is the use of the symbolic mathematical language to describe
mathematically the phenomena (usually) expressed in natural language.
We employ both languages, the one that describes our world in natural language,
and the one that attends to it in a formal language. Yet, it is important
to keep in mind that these are two distinct languages.**

**Natural language is limited in its ability to describe
mathematical notions. Take the following trivial example: write down the
expression "a fourth of a number decreased by five". It looks to be part
of natural language; at least it is not written in a symbolic manner. This
is, of course, an ambiguous expression. It can also be written, either
as 1/4 (N - 5) or as 1/4 N - 5.**

**From a mathematical point of view, these expressions
have different meanings. However, it is only with formal notation
that we can differentiate between them. To talk mathematically means in
part to be able to formulate expressions in a way that will distinguish
between the two meanings, and this cannot be accomplished without formal
notation. Mathematicians, by agreeing about the role of parentheses in
its formal notation, are able to make such distinctions, whereas the users
of natural language must have recourse to intonation, which is missing
in the written form of natural language.**

**There are many terms in mathematics and science which
do not exist or have a different meaning in natural language. Here are
a few examples:**

**We use natural language to speak about mathematical
notions. We listen to students’ talk in order to evaluate what they have
understood. Learning of the square root as a mathematical notion, however,
will be evaluated by the learners’ ability to apply this notion in various
contexts that call for the use of this formal concept. In other words,
students will be evaluated according to their ability to talk mathematically
in using this term, and not by the way they talk about it in natural
language.**

**While acknowledging that it is important to talk with
children about their formal mathematical activities and that explaining
their own actions may, indeed, support their learning of mathematics, we
should keep in mind the main issue: that talking mathematically, means,
first of all, acquiring the ability to describe the world’s situations
with the formal models of mathematics.**

**This brings me to the neo-pragmatist argument. What
does it mean that knowledge is socially derived? A mathematician’s creation
has a social context. The mathematician does not work in a vacuum, even
when he works alone. The acceptance of his creations as a part of ‘standard
mathematics’ has, of course, a social aspect to it. Yet, what we teach
at school are the standard notions and methods of doing mathematics. Therefore,
we must accept the universality of mathematics as a language in which to
talk about events detached from contexts. We usually ground the mathematical
knowledge in various contexts and we apply it in real-life situations which
we first describe in everyday language. We talk also about mathematical
activities in natural language, which would then serve as a meta-language.
Yet, in essence, mathematics is a formal language which is helpful because
of its universality and due to its special syntax and semantics. This is
the understanding of mathematics which guides our teaching.**

**Thus, although both teaching and creating mathematics
take place in social contexts (in different ways), the most essential characteristic
of what mathematicians are creating (and what we are demanding our students
learn) is its universality and independence of context. To these two features
mathematics owes its strength as a tool of communication.**

**I see many cases in which conversations can help: in
realising the difficulties children encounter, in learning the variety
of applications, in trying to clarify to oneself the thinking process,
etc., and this brings me to the interactionist argument. I would like to
differentiate among:**

**2. Conversations between a teacher and student
that serve many purposes. From the constructivist point of view, it is
the major means by which the teacher has the opportunity to learn about
the student’s thinking and have a real dialogue.**

**3. Conversation among children in which the
students try to explain their methods. In a way, this is parallel to the
mathematicians’ conversation. In the process of education we could benefit
from this kind of conversation, if (and this is a big IF) the rules
of the game were clear. The teachers, as a part of their role, could help
the students learn what is a convincing argument in mathematics and how
it is similar to, or different from, an ethical or artistic argument.**

**C. The conversation of mathematicians versus students’
mathematical discussions**

**Leen Streefland**

**In order to illustrate my position with respect to
the focal question, I have selected three examples. Two of them come from
the history of mathematics and one is taken from classroom observation.
Several basic conditions need to be fulfilled if mathematical communication
is to take place. There is also the question of basic assumptions. One
such may be as follows:**

**In the Acta Eruditorum of June 1696 (ill. p.
269), Johann Bernoulli issued "an Invitation to all mathematicians to solve
a new problem" (one which he called the brachystochrone problem – in Greek,
brachus means ‘short’ and chronos ‘time’).**

**Bernoulli assured his audience that applications of
the solution of this problem would go well beyond mechanics. At the end,
he warned his colleagues not to judge too fast. By saying this he excluded
as a possible solution the straight line from A to B, which, in spite of
its being the shortest path, would not require the shortest time. He also
promised that by the end of that year he would mention the curve AMB in
his solution and added that this curve was well known to geometrists (van
Maanen, 1995). On June 9, 1696, Bernoulli sent the problem to Leibniz,
who answered already on June 16. He suggested ‘Stachystoptote’ – the ‘line
of fastest descent’ – as the name of the required curve. He also said the
problem was wonderful and added: "Although I resisted, it attracted me
like the apple attracted Eve".**

**Bernoulli published his solution in the Acta Eruditorum
of May 1697. I will only mention one of his ideas – the one which was deemed
as brilliant by other mathematicians. He imagined that the problem was
not about a ball or a point of mass rolling down, but about a ray of light
shining through a material with variable density. According to Fermat’s
principle, as referred to by Bernoulli in his solution, light always takes
the path which ensures the shortest transition time. Therefore, the ray
would follow the brachystochrone if it passed a medium in which at height
x it would have exactly the same transit velocity as the known rate
of fall of the little ball at height x.**

**Bernoulli sent his solution to Leibniz already on July
21, 1696 and Leibniz reacted to it on July 31. He wrote, among others:**

**Andrew Wiles worked on the proof of Fermat’s last theorem
for seven years. As Wiles himself told us in the television documentary
on his achievement, he had had a passion for Fermat’s problem since his
adolescence. For him, it was "a beautiful problem, a great challenge".
"Mathematicians just love challenge. It was a my private battle", he said.**

**In this project, Wiles went through several stages.
Working alone in complete isolation was the first phase (even admitting
to working on Fermat’s last theorem would certainly have raised some eyebrows!).
Wiles decided to transfer elliptical curves into Galois representations
in order to make them countable. This step took three years. Later, Wiles
hoped to make his counting strategy complete by applying the Iwasawa theory.
[3] Then he skipped from here to the class number theory (formulas of Flach
and Kolyvagin). All this took seven years. It was only then that he eventually
presented his findings at a conference, after talking to one or two colleagues,**

**Before publication, his colleague Nick Katz went through
the manuscript and emailed Wiles once or twice daily with questions such
as "What do you mean by this?" or "What does that mean?". It turned out
that Wiles’ line of thought contained a mistake, which was discovered by
Katz. Several mathematicians, such Richard Taylor, a former student of
Wiles, came to his assistance in an attempt to overcome the difficulty.
The proof was eventually completed with the help of Taniyama–Shimura conjecture,
which had already played a role at an earlier stage of the Wiles’ work.**

**John Mason has made the following observation of what
mathematicians are usually doing.**

**To sum up, mathematical productivity, communication
and meta-cognitive reflection go hand-in-hand and seem to constitute an
all-important triad. As far as I am concerned, mathematical communication
owes its significance to the co-existence of mathematical production and
reflection. This has already been illustrated by the example of communication
between Johann Bernoulli and Leibniz on the brachystochrone problem. Notice
that proving something does not mean making sure that one’s thinking is
true beyond doubt; rather, it shows that proving means, first and foremost,
increasing coherence by connecting and integrating different mathematical
ideas and theories.**

*Third example: Impression from a mathematics classroom
– measuring the height of a tower*

**Some sixth graders estimated the height of a tower
by comparing it with objects of reference like a door, the height of a
van, the estimated length of a segment of the tower, and so on. They did
so with the help of a photograph of the tower. The two teachers simultaneously
present in the classroom invited the children both to think about the problem
and to compare their methods. One teacher hinted that on a sunny day there
would be a special method to think up.**

**The children worked in small groups first, and then
there was a classroom discussion. It can be ascertained from the protocols
that there were only two small groups of children who took the sun-and-shadow
approach seriously, with boys only in both groups. When during the whole-class
discussion the teachers invited the children to propose their solutions,
Micha was the first one to come up with the idea using a sun-and-shadow
approach. His reasoning boiled down to the following comment:**

**There was a repeated intervention by Bart, who seemed
to have reached a global perspective on the sun-and-shadow question. He
claimed that a vertical object and its shadow must be equal twice during
one day; however, the boy did not succeed in making his point in a comprehensible
manner, and, as a result, his input initially did not influence the discussion.**

**Then, there was a contribution from another boy, Peter,
who during the group work period acted like a little Andrew Wiles in that
at a certain stage he stopped listening to what the different members of
his group were saying and began to sail his own course. Now, he may have
saved the situation by supporting Micha’s suggestion.**

**After some elucidations and additional remarks from
others, Micha voiced his objections saying that the tower is a monument,
and therefore "You are not supposed to nail a lath to it". At the same
time, however, he had to admit that Peter’s idea worked nicely. Saskia
proposed a compromise: why don’t you use your own shadow? Bart intervened
again and suggests that Peter’s little lath can also be used for his own
idea, namely to produce a shadow at a distance of two metres from the foot
of the tower. Finally, Micha blew the whole theory by asking:**

**This brief classroom observation, which has been presented
here only in general terms, reveals that:**

**• this classroom interaction brings about different
levels of insight, ranging from ones having to do with some particular,
local phenomenon of the sun-and-shadow question right up to those which
deal with a global, overall perspective;**

**• creating a climate which encourages broad participation
also can provoke reflection in students who did not deal with the given
question before (compare this with the finishing touch of Andrew Wiles’
proof!);**

**• some pupils behave like little mathematicians: like
Wiles, they would initially work in a solitary manner, and later they would
make creative contributions which would allow others to join in whenever
the break-through ideas resonated with their own experiences. [4]**

**To sum up, classroom discussion provokes a lot of reflection
and gives an opportunity to compare, criticise, refute, complete, reject,
and so on. Meta-cognitive shifts are commonplace in such process (compare
the historical examples in Streefland, 1996). There is a triad of principles
which must be observed if the discourse is to be effective. The participants
have to**

**(b) communicate mathematically in a productive manner;**

**(c) allow for meta-cognitive shifts.**

**D. Theorizing about mathematical conversation versus
learning from practice**

**Paul Cobb [5]**

**My immediate response to Sfard’s question is to ask
what the alternative might be, given that conversations occur in all classrooms.
For example, Japanese elementary mathematics teaching is often held up
as an exemplary form of instructional practice in which students are encouraged
to engage in discussions by explaining and justifying their reasoning (Stigler,
Fernandez, and Yoshida , 1996). However, when the number of words spoken
per minute is taken as a crude measure of the amount of talk occurring
in classrooms, the results of the recent TIMSS video-study indicate that
there is actually more discourse in traditional American elementary classrooms
than in Japanese classrooms (Stigler, personal communication). This finding
conflicts with our expectations in that traditional American classrooms
are often portrayed as being devoid of discourse.**

**The emphasis placed on discourse in recent reform recommendations
(such as the NCTM Standards) in fact represents a reaction to this
commonly-accepted view of traditional American instructional practices.
On my reading, the major point of the TIMSS finding is that although there
is more discourse in American classrooms, less of mathematical significance
is being said. I therefore suggest that the question is not whether students
should engage in conversations, but instead concerns the nature of those
conversations that constitute productive situations for mathematical learning.**

**In each of the three arguments that Sfard rightly challenges,
proponents of conversation attempt to derive instructional implications
directly from general, orienting, background theories. In other words,
the implicit strategy in each argument is to translate basic theoretical
tenets into instructional prescriptions. In my view, this rhetorical move
involves a basic category error in that the three theoretical perspectives
are descriptive rather than prescriptive. Their basic tenets
are concerned with how to interpret human activity and, as a consequence,
they simply do not give rise to instructional implications.**

**The classic example of this type of category error
arises when the theoretical commitment to analyze mathematical learning
as a constructive process is translated into the instructional prescription
that teachers should enable students to construct mathematical understandings
for themselves by not telling them anything. Constructivism does not, however,
have a monopoly on this type of error. It is also prevalent in discussions
of distributed theories of intelligence (as intelligence is distributed,
computers and other tools should always be available for students to distribute
intelligence over them) and situated learning theories (as mathematical
learning is situated, instructional situations should always involve authentic,
real-world problems).**

**I might also note that attempts to derive recommendations
for teaching directly from any philosophy of mathematics or science is
similarly flawed. In my view, many of the questionable pronouncements that
pass for instructional theory reflect this error, thereby subjugating the
wisdom of practice to the fervor of ideological commitment.**

**If we do not rely on general background theories to
tell us how to teach, how then might we approach the question posed? I
am enough of an empiricist to suggest that we experiment in classrooms,
that we look at the conversations that do arise, that we focus in particular
on the nature of the students’ participation in those conversations, and
that we investigate what students might actually be learning in the course
of that participation.**

**I would readily acknowledge that this way of framing
the question reflects my basic theoretical commitments. My focus on both
participation and students’ learning indicates that I subscribe to a social
constructivist perspective. However, it is one thing for general theoretical
commitments to orient how one casts questions, and another for these commitments
to serve as the primary touchstone for instructional decisions.**

**I would also note that the way I have framed the question
is unusual (at least in the American context) in that it attends to what
students might actually be learning. Frequently, a particular vision of
classroom discourse is held up as the ideal and instructional interventions
are judged to be successful if actual classroom discourse approaches this
ideal. In these interventions, shaping classroom discourse has become an
end in itself, and the issue of whether students might be learning any
mathematics that is worth knowing is not a focus of investigation. In my
view, the justification for any intervention including those that involve
discourse has to attend to the changing nature of students’ participation
in the practices established by the classroom community and thus to their
mathematical learning.**

**I will illustrate the general approach that I have
outlined by describing two aspects of classroom conversations that I and
my colleagues have found to be potentially productive for students’ mathematical
learning. I should stress that these features of conversations were not
derived a priori from theoretical principles but have instead emerged
in the course of a series of classroom teaching experiments conducted in
collaboration with teachers over a ten-year period.**

**The first aspect draws on the work of Thompson and
Thompson (1996) and concerns a distinction that they make between calculational
and conceptual orientations in teaching. We have found it useful to extend
this distinction by talking of calculational and conceptual discourse.
It is important to clarify that calculational discourse does not refer
to conversations that focus on the procedural manipulation of conventional
symbols that do not necessary signify anything for students. Instead, calculational
discourse refers to discussions in which the primary topic of conversation
is any type of calculational process. This can be contrasted with conceptual
discourse in which the reasons for calculating in particular ways can also
become explicit topics of conversation. In this latter case, conversations
encompass both students’ calculational processes and the task interpretations
that underlie those ways of calculating.**

**In our experience, classroom discussions in which the
teacher judiciously supports students’ attempts to articulate their task
interpretations can be extremely productive settings for mathematical learning.
In general, we have found that the development of classroom discourse and
the development of ways of symbolizing and notating go hand in hand and
are almost inseparable. I would also note that in helping students explain
their task interpretations, the teacher is simultaneously initiating and
guiding the renegotiation of the socio-mathematical norm of what counts
as an acceptable explanation. We have found that, within a few weeks, most
students routinely give conceptual explanations as the need arises and
that they ask others clarifying questions that bear directly on their underlying
task interpretations.**

**The second aspect of discourse that we have found to
be productive for students’ mathematical learning builds on the first and
involves what we call reflective shifts in discourse. These shifts
occur in classroom discourse when what was said or done in action subsequently
becomes an explicit topic of conversation. A shift of this type can occur
when the teacher and students compared and contrasted task interpretations.
For example, in a class, initially, the task was simply to solve the word
problem and the students reported their solutions. Later, these solutions
and the underlying interpretations became topics of conversation. What
had previously been said and done in action became a topic of conversation
in that the theme of the discourse was now the structural characteristics
of various solutions. I and my colleagues have argued elsewhere that in
making shifts of this type, the classroom community engages in a collective
act of reflection (Cobb, Boufi, McClain, and Whitenack, 1997).**

**In explaining why reflective shifts in discourse can
be productive for students’ mathematical learning, I draw heavily on Sfard’s
(1991) theory of reification. Briefly, she accounts for mathematical development
in psychological terms by contending that students’ initial, action-based
operational conceptions evolve into object-like structural conceptions
via a process of reification. When we analyze instances of reflective shifts
in discourse, we see this same process occurring in the collective activity
of the classroom community. Consequently, we hypothesize that opportunities
for students to reflect on, objectify, and reorganize prior mathematical
activity arise as they participate in reflective shifts in discourse.**

**In making this conjecture, we note that students do
not just all spontaneously happen to begin reflecting at the same moment.
Instead, their reflection is enabled by their participation in the discourse,
and by reflecting they contribute to the shift in discourse. Further, when
viewed in broader terms, conceptual discourse that involves reflective
shifts is consistent with Streefland’s observation that discourse needs
to be both mathematically constructive and productive.**

**In concluding this brief discussion of classroom discourse,
I should clarify that when I and my colleagues first began working intensively
in classrooms ten years ago, we required students always to work
in small groups and then to participate in whole-class discussions. Thus,
if students in traditional classrooms were not allowed to explain how they
actually interpreted and solved tasks, the students in these classrooms
were not allowed to shut up and engage in mathematical activity on their
own.**

**Over the years, we have modified the classroom activity
structure in several ways. For example, the students now often work individually
but on the understanding that they can talk with peers of their choosing
as the need arises. One of the teacher’s responsibilities is then to help
the students learn how to use their peers effectively as resources for
their learning. In addition, we do not organize a whole-class discussion
of students’ individual or small-group work, unless we anticipate
that mathematically significant issues that will advance our pedagogical
agenda might emerge as explicit topics of conversation.**

**Thus, whereas we were previously satisfied if students
engaged in a discussion that appeared to have a mathematical theme, we
now consciously plan discussions by first monitoring the various ways in
which students are interpreting and solving tasks. We then often select
particular students to call on because we conjecture that a potentially
significant issue might emerge with the teacher’s guidance either when
discussing a particular solution, or when comparing two or more solutions.**

**In taking this approach, we no longer value discourse
for its own sake. Instead, each classroom discussion, viewed as social
event in which students participate, has to be justified in terms of whether
the issues that emerge contribute to the achievement of our potentially-revisable
goals for students’ mathematical learning.**

**Ironically, the discussions conducted in the classrooms
in which we worked ten years ago often appear to the untrained eye to be
superior in that they are characterized by a hubbub of activity. However,
when we focus on the students’ participation and what they are learning,
the discussions conducted in recent classroom teaching experiments are
strikingly superior to those in the first classrooms in which we worked.
As was the case with the comparison of the Japanese and American classrooms,
more that is mathematically significant is being said, even though there
is less talk.**

**F Talking while learning mathematics versus learning
the art of mathematical conversation**

**John Mason**

**The term ‘conversation’ is sometimes taken to cover
any verbal interchange, especially social interchange, between people.
‘Conversation’ has also been taken to refer to the entire interaction between
the individual and the social, whether in the form of other people or with
the culture (through acting and interacting) or more particularly, the
historical detritus of previous ‘conversations’ in the form of genres of
text and patterns of discourse. Thus, Maturana (1988) sees conversation
as the way of being-in-the-world.**

**‘Discussion’ is usually taken to mean a more focused
conversation. For example, in mathematics, Pirie and Schwarzenberger (1989)
have suggested that it requires four elements: to be purposeful, on a mathematical
topic, with genuine pupil involvement, and interactive. They were trying
to circumvent the classic teacher report ‘we discussed’, simply meaning
that the teacher expounded. At issue for most practitioners is achieving
productive mathematical discussion: one common teacher fear is that students
will chat about other things, and in some classrooms, as well as mathematical
discussion, there is a great deal of unfocused or off-task interaction
which may or may not be seen as productive and conducive to learning mathematics.
Note, though, that adults in the work-place also talk about a variety of
topics, because personal communication is vital to effective task communication
in a non-authoritarian environment – but like anything, it can be overdone.**

**Fundamental to productive mathematical conversation
and conversation about mathematics is the development of a conjecturing
atmosphere (Mason et al., 1984; see also Baird and Mitchell 1986,
and Baird and Northfield 1992). In a conjecturing atmosphere, everything
that is said is said as a conjecture, uttered with the intention of externalising
thoughts so as to be able to examine them critically, and to modify them,
often as a result of other people’s comments. This is by way of contrast
with an atmosphere in which utterances are expected to be pre-formulated,
correct, and justifiable. In a conjecturing atmosphere, there is a shared
struggle to find ways to express and convince which can be understood and
appreciated by others, as well as challenged, exemplified, amplified, varied,
generalised, etc. by them.**

**Some people seem to describe conversation as though
it were the principal or even the sole means whereby students (re-)construct
ideas; that collectively students can (re-)discover the essential features
of mathematics. But this makes no sense in any reasonable perspective,
as without encounters with the ideas of others, whether from the historical
past or the relatively-expert present, most students are unlikely to reconstruct
the important ideas of mathematics. They need to be in the presence of
more highly structured awarenesses, in the form of carefully constructed
tasks, exposition, and people. Or as Vygotsky (1981) put it, in the presence
of ‘higher psychological processes’, which they then reconstruct and adapt.**

**Thus, a critical component of effective discussion
is the presence (possibly virtual) of a relative expert. This presence
is not mere physical presence but rather the presence of awareness of mathematical
thinking processes as well as content. For example, a group of dedicated
and disciplined students can have fruitful discussions based on reading
exposition by an author; the effectiveness of their discussions will be
determined by the way of working which the group develops, including mutual
support, individual and group intention, as well as group discipline.**

**The social constructivist-based argument that since
mathematics is a discourse, students will somehow learn mathematics through
developing a discourse amongst themselves, is as specious as the behaviourist-based
argument that students will learn mathematics by being conditioned or trained
to employ techniques and algorithms correctly on typical test items.**

**Conversation as a mode of interaction**

**Conversation is a form of interaction. In Mason (1979),
I elaborated how action is essentially triadic in quality, involving affirming,
mediating, and responsive impulses, based on ideas of Bennett (1966). I
label the six actions which arise when the tutor (standing for a relative
expert), student, and content take up those three roles in all possible
ways. The result is what I call the six Ex’s, with names chosen because
they come close to capturing the essential quality of the respective modes
– but I note that these words are being used in technical ways. Each triangle
can be ‘read’ using the formula ‘the affirming acts upon or contacts
the response, facilitated or enabled by the mediator’. In
the diagram below, the affirming is at the top, mediating in the middle,
and responding at the bottom.**

**Insert Fig. 3 approximately here**

**It is easy to generate the six diagrams mathematically,
but more difficult to make sense of them. In particular, some traditional
meanings of these words have to be amplified or modified in order to capture
the quality of the corresponding mode.**

**Briefly summarised, the six modes are as follows:**

*Expounding***Expounding is what produces exposition, the form which
is most familiar in texts. True exposition occurs when the presence (actual
or virtual) of the audience enables the expounder to make contact with
the content in fresh and vital ways. The experience of preparing a session,
with a flood of rich connections amongst which choices have to made, is
typical of the energy of this action. The audience is attracted into the
world of experience of the expounder, whose contact is heightened by the
presence of the audience. The expounder could be said to be in conversation
with the content, enabled by the students’ presence.**

*Explaining***True explaining comes about when the tutor or relative
expert tries to enter the world of the student, by means of, through, and
concerning, the content. As soon as ‘the difficulty’ becomes clear, the
temptation is very strong to slip into expository mode. To an observer,
tutor and student may be in ‘mere’ conversation, but the energy and attention
is highly focused on the student’s experience.**

*Exploring***The student takes the initiative and makes contact
with the content through the presence and guidance of the tutor. The student
might be reading a text and actively making sense (re-constructing, re-specialising,
re-generalising, conjecturing, re-justifying) guided by the text, or by
materials designed by a relative expert. In a sense, the student is in
conversation with the content, mediated by a virtual or actual relative
expert manifested in person or as structured materials, text, etc.**

*Examining***The student submits to the relative expert in order
to validate their own judgement that they understand and have re-constructed
appropriately. This is different in energy from the degenerate form of
assessment in which students are tested because someone else has decided
the time and place independently of whether the student is ready or not.
It is vital that students validate their own criteria against those of
the expert, rather than simply submitting to being tested like a quality-control
check on a production line. In some forms of examining, such as oral testing,
a conversation could be said to be going on, but the nature and focus of
that conversation is assessment orientated.**

*Expressing***In expressing, the content bursts forth spontaneously,
often unstoppably, using the presence of an audience to release energy
of connection-making and understanding, however conjectural. This is characteristic
of brainstorming, and of the triggering of idiosyncratic connections which
marks social conversation, but which also plays an important part in working
on mathematics. It is almost as if the content is using the student in
order to participate in conversation!**

*Exercising***Here the content draws the student into rehearsal
to mastery, as a child will repeat certain actions over and over until
they are automatic. Note the contrast from the degenerate form in which
the tutor urges exercises upon the student who then completes them with
a minimum of energy and attention. This action could be seen as a conversation
between the student’s awareness and some of their functioning selves as
they integrate skills and techniques into their soma.**

**Problematicity**

**For me, the various modes of interaction are mutually
supportive, so that personal and collective work prepares the ground for
hearing what an expert has to say, and hearing what an expert has to say
prepares the ground for personal and collective work. Engaging students
in mathematical conversation and in conversation about mathematics is not
the issue, but there are some significant issues which do arise.**

*Intention***Education is not something that is done to people;
it is an action leading to activity in which people participate in different
ways and with different intensities. This view is consistent with Leontiev
(1981), who distinguished three levels of activity: energised or motivated
activity, action defined by a goal, and operational activity. Operational
activity is unaware of goals or subgoals, and energised or motivated activity
may not have conscious goals.**

**Thus, conversation–discussion is not in itself valuable.
Rather, its value at any time depends on the role played in the situation,
and the commitment of individuals and of the group collectively.**

*Problematicity of changing modes***Getting meaningful discussion started is not necessarily
difficult in itself. The technique of talking in pairs is an excellent
way to start, having students rephrase something for themselves to each
other before inviting them to contribute to a more plenary discussion.
The opportunity to try something out in relative safety, discovering that
what you think is indeed worth saying and not ridiculous, supports more
sophisticated contributions to discussions. Once established, a conjecturing
atmosphere as described earlier supports people in exposing their uncertainties
as conjectures which can then be worked on, modified, and developed without
fear of ridicule or unpleasant exposure.**

**Moving into individual work is just as important as
collective work, in order to allow individuals to reconstruct ideas, situations,
and techniques for themselves. Merely being present when someone else does
something can be sufficient for some people in some circumstances, but
usually individuals need to spend time re-visiting and re-constructing,
so that they can hope to re-construct again in the future when it is required.**

**Most difficult is moving from individual work to collective
work: listening, adapting to and building on others’ thinking, learning
to suppress one’s own approach in order to appreciate someone else’s, learning
to express one’s own approach in ways which others can enter and appreciate.**

**Acquainting students with the different modes, their
requirements and their effects helps educate their awareness and so help
them to learn from experience.**

**F. Believing in learning through conversation is not
enough: a conclusion**

**Anna Sfard**

**In spite of Cobb’s and Mason’s sober reminder that
only experience can bring an ultimate verification of the claims on the
pedagogical advantages of conversation, I feel a summary urge to make a
theoretical comment. I wish to admit now my own faith in the power of conversation
and to stress that this faith is not an isolated opinion, but rather a
matter of a world-view according to which all our thinking, with
mathematical thinking being no exception, is essentially discursive. [6]**

**Like interactionists and neo-pragmatists, and probably
like most of the panelists in this group, I view our conceptual systems,
and thus both our human selves and the world each one of us lives in, as
created through and within the activity of speaking, whether public or
private. Being discursive creatures, we cannot simply step out of the discourse.
Discourse is where all our cognitive acivities start, exist, and come to
closure.**

**As such, all these activities are essentially social,
and even their occasional appearance as leading to universal and mind-independent
results is but a discursive by-product. If so, the more aware we are of
the discursive processes that constitute our mathematical activity, the
better chance we have of attaining appropriate control of these processes;
the better our control, the more effective our students’ learning. In short,
the question is not whether to teach through conversation, but rather
how. Since learning mathematics may be equated to the process of
entering into a certain well-defined type of discourse, we should give
much thought to the ways students’ participation in this special type of
conversation might be enhanced.**

**And when it comes to directing and orchestrating a
helpful exchange, all the panel participants seem to agree on the decisive
role of the teacher (or any other ‘relative expert’). To a great extent,
it is up to him or her whether a given mathematical conversation, designed
for the purpose of learning, will be a success or a failure. There are
many ways to turn classroom discussion or group work into a great supplier
of learning opportunities; there are even more ways to turn them into a
waste if time, or worse than that – into a barrier to learning.**

**As it happens, futile, useless, and even potentially
harmful types of discursive activities can be observed only too often in
mathematics classrooms all over the world. One such case, now being described
by Carolyn Kieran and myself (Sfard and Kieran, 1997), comes from our own
teaching experiment carried a few years ago in Montreal. Two reasons may
be responsible for this state of affairs. First, to charge the teachers
with the responsibility for the effectiveness of the conversation is easy,
but to support them in attaining this goal with helpful advice is difficult.
Orchestrating a productive mathematical discussion or initiating a genuine
exchange between children working in groups turns out to be an extremely
demanding and intricate task.**

**Second, our Montreal experience has shown with particular
clarity what psychologists have long known: communication skills cannot
be taken for granted. Children are not necessarily born with a natural
talent for transferring their intentions to others or with knowledge of
how to interact with others to enhance everybody’s understanding. If conversation
is to be effective and conducive for learning, the art of communicating
has to be taught. How this is to be done, and what exactly should be learned
by the students remains a question to which the mathematics eduction community
has yet to give much thought.**

**Perhaps all one can say right now is that for a conversation
to be productive, it has to have the characteristics of a true dialogue.
For the top-level description of the task, it may be useful to consult
Gadamer (1975), according to whom a true dialogue:**

**Notes**

**1. The debate originally took place orally in June
1997 in Calgary, Canada, at the Third International Conference of History,
Philosophy and Science Teaching. The panel was chaired by Anna Sfard, who
also edited this written version.**

**2. The shift of the academic focus toward the social
and away from the pyschological has played an important contributory role.**

**3. During this period, Andrew Wiles walked a lot along
a lake, where it was quiet and peaceful. According to his own account,
it was good to relax there, to concentrate and to let the ‘subconscious
mind’ work.**

**4. The loneliness of a top mathematician is present
in the position of Bart, whose initial insights are beyond those of the
rest of the group.**

**5. The analysis reported here was supported by the
Office of Educational Research and Improvement (OERI) under grant number
R305A60007. The opinions expressed do not necessarily reflect the views
of OERI.**

**6. See Edwards and Potter, 1992; Harré and Gillet,
1995.**

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