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Pause (2 secs)
Teacher: Zena asked a question.
[Chorus of yesses from the class.]
In relation to my earlier mention of Ainley’s work on questioning, I am led
to wonder whether Zena appreciated any difference between her two ques-
tions that were dealt with very differently by the teacher. Here, his meta-
comment "Zena asked a question" offers a deflection that allows him appar-
ently to take a turn in the conversation yet without having to respond to
Zena’s request for evaluation directly.

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4.2 Modality and Hedges
1
A second general area arises under the general heading of "modality,"
which initially referred to the use of modal verbs (see Stubbs, 1986) to mark
the degree of speaker certainty or uncertainty (e.g., "that might be true"), but
now has a more general meaning. One discussion of the notion in relation to
mathematics learning can be found in Anne Chapman’s (1993) doctoral dis-
sertation Language practices in school mathematics: A social semiotic per-
spective. She writes:
Hodge and Kress (1988) use the semiotic term modality to describe the social
construction or contestation of knowledge. Modality refers to the degree of cer-
tainty embedded in a statement.... In any school subject, the weighting attached
to what is said is important. Mathematics, in particular, is typically regarded as a
factual subject and thus is likely to have a high modality structure. (p. 57)
What other linguistic means are commonly available and used in mathemat-
ics classrooms for indicating the speaker’s relation to or stance taken with
respect to some knowledge claim uttered? In John Wyndham’s novel The
Kraken Wakes, for instance, one of the characters reports:
1
A general term in this area is "hedge" (see, e.g., Lakoff, 1972), though Prince, Frader, &
Bosk (1982) have usefully distinguished between "hedges" and "shields." An example of a
shield is "I think that X is true," where the uncertainty is in relation to the speaker’s level
of confidence in the truth of the assertion, while a hedge, such as "the cost is approximately
£20," has the uncertainty marker inside the proposition itself.
166
MATHEMATICS CLASSROOM LANGUAGE
A teacher and a student are putting up posters and having to take out many old
staples:
Student: Do we have to take them all out?
Teacher. You can sweep dust under the carpet too.
Pragmatics is an area of linguistics dealing with how words can be used
to do things, to achieve one’s ends. The philosopher Paul Grice (1989) has
proposed a co-operative principle and a series of very general maxims to try
to account for how and why discourse works and coheres. He cites the ex-
ample of the book review, which, in its entirety, runs: "This book has nar-
row margins and small type." What implicatures must be made in order to
construe this as a book review? One of Grice’s suggestions enjoins us to be-
have so as to "avoid obscurity of expression, avoid ambiguity."
1. The maxim of Quality (be truthful, according to the evidence you have).
2. The maxim of Quantity (be informative, but not over-informative).
3. The maxim of Relevance (be relevant to the conversation).
4. The maxim of Manner (say things clearly, unambiguously, briefly).
I have yet to look at the notion of meta-commenting in relation to viola-
tions of Grice’s maxims. But it is an interesting observation that many of
Grice’s maxims of conversation are regularly and systematically violated in
classroom discourse.

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DAVI
D PIMM
Seeing how the status of and beliefs about the validity of knowledge claims
are crucial in mathematics, again it seems curious to me that more is not
known about how these pragmatic utterances are made. Though it must be
said this forms a subtle part of communicative competence. Recently, a
similar shift of focus and concern has occurred in mathematics education to
that from syntactic to semantic and then to the burgeoning area of pragmatic
issues present in linguistics itself. I predict the extremely subtle pragmatic
interpretative judgements regularly made by both teachers and students in
the course of mathematics teaching and learning in classrooms will move
steadily to the fore as a research topic.
4.3 Force
My current thesis is quite simple. All that hearers have direct access to in
the classroom is the form of any utterance. But that form is influenced and
shaped by the intended function of the utterance (some particular examples
of general teacher functions include: keeping in touch, to attract or hold stu-
dent attention, to get them to speak or be quiet, to be more precise in what
they say). And form is also shaped by personal force, the inner purposes and
intentions of the speaker, usually in this case what the teacher is about both
as a teacher and a human being.
I am currently exploring some aspects of mathematics classroom dis-
course with regard to:
1. Linguistic form (all that is actually readily available to the external ear
and eye): for instance, pronominal usage and deixis (Pimm, 1987, on "we";
Rowland, 1992, on "it"). Mathematics has a problem with its referents, so
the ways in which language is made to point is of particular interest.
2. Some of the apparent or hoped-for functions (quite common and gen-
eral ones, such as, for the teacher, having students say more or less, deflect-
ing questions; or for the student, avoiding exposure, engaging with the con-
tent, finding out what is going on).
3. Force. The personal, individual intents (conscious and unconscious)
that give rise to the desire to speak. I start from the premise (that of Anna
Lee, founder of the Shakers) that "Every force evolves a form."
I believe that force and function combine to shape form, but also that the
existence of conventional forms of speaking, the pressure of certain class-
room discourse patterns, can actually interfere with expression. I am also
becoming increasingly interested in how the notion of force, of necessity
must include "unconscious force." (See Blanchard-Laville, 1991,1992, for a
167
"For present purposes the danger area is being reckoned as anything over four
thousand", said Dr Matet . . . .
"And what depth did you advise as marking the danger area, Doctor?"
"How do you know I did not advise four thousand fathoms, Mrs Watson?"
"Use of the passive, Doctor Matet – ‘is being reckoned’." . . .
"And there are people who claim that French is the subtle language," he said.
(Wyndham, 1970, pp. 101-102)

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MATHEMATICS CLASSROOM LANGUAGE
REFERENCES
Aiken, L. (1972). Language factors in learning mathematics. Review of Educational