When the Rules of Discourse Change, but Nobody Tells You: Making Sense of Mathematics Learning From a Commognitive Standpoint
Abstract
The interpretive framework for the study of learning introduced in this article and called commognitive is grounded in the assumption that thinking is a form of communication and that learning mathematics is tantamount to modifying and extending one's discourse. These basic tenets lead to the conclusion that substantial discursive change, rather than being necessitated by an extradiscursive reality, is spurred by commognitive conflict, that is, by the situation that arises whenever different interlocutors are acting according to differing discursive rules. The framework is applied in 2 studies, one of them featuring a class learning about negative numbers and the other focusing on 2 first graders learning about triangles and quadrilaterals. In both cases, the analysis of data is guided by questions about (a) features of the new mathematical discourse that set it apart from the mathematical discourse in which the students were conversant when the learning began; (b) students' and teachers' efforts toward the necessary discursive transformation; and (c) effects of the learning–teaching process, that is, the extent of discursive change actually resulting from these efforts. One of the claims corroborated by the findings is that school learning requires an active lead of an experienced interlocutor and needs to be fueled by a learning-teaching agreement between the interlocutor and the learners.
When the rules of discourse change, but nobody tells you
- the case of a class learning about negative numbers
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Address for correspondence: Anna Sfard, 45 Ha-Eyal Str., apt. 3, Jerusalem 96955, ISRAEL, e-mail:
sfard@netvision.net.il, fax +972-2-6796899
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Episode 1: The first lesson on negative numbers
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1. What does it mean to learn mathematics? The communicational answer.
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Some writers go on to question even the ostensibly unproblematic case of the talk about concrete objects.
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Our use of the term routine is close to the usage that has been proposed by Schutz & Luckmann (1973) and
applied in the context of mathematics learning by Voigt (1985). The notions social norms and
sociomathematical norms, introduced by Cobb, Yackel, and their colleagues (see e.g. Yackel & Cobb, 1996),
although not tantamount to the idea of meta-discursive rule (not every meta-rule is a norm; see Sfard 2000b),
are clearly related to the same phenomena.
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2. Focus on the discourse (mathematics): What is to change in the endorsement routines
when negative numbers are introduced?
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On the face of it, this claim may be contested since many ideas have been proposed to model negative
numbers (e.g. there is the model of movement where time, velocity and distance can be measured in negative
as well as positive numbers; numbers may be represented as vectors, etc.). And yet, at the closer look, all of
these explanations turn out to be derivatives of the same basic decisions about preserving certain former rules
of numbers while giving up some others; these fundamental choices are exactly the same as the ones that find
their expression in the acceptance of axioms of numerical field as a basis for any further decision, and they
must be (possibly in a tacit way) accepted prior to any justification (see also Sfard 2000b).
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3. Focus on learning interaction: What do the teacher and the students do to make the
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The Hebrew expression “ve-od,” which literally means “and [one] more,” is used in school mathematics in
the sense of “add” or “plus” (the word plus itself may also be used).
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Note that the teacher’s argument is a school version of the one that has been presented formally in
Figure 1: In both cases the laws of multiplying signed numbers are derived from the laws that hold for
unsigned numbers according to the principle of preserving certain basic features of numerical operations.
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4. Focus on effects: Did the expected change in endorsement routine occur?
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6. Discussion and conclusions: On communicational conflict and realistic communicational
agreement as the necessary conditions for learning
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8
Let me remark that communicational conflict is often involved also in mathematical invention (or any other
scientific invention, for that matter). In this case, the conflict is likely to occur within a person: in the
transition from a familiar discourse to a new one the mathematician may find himself endorsing conflicting
narratives. One of a well-known cases of such inner conflict is that of George Cantor, the inventor of set
theory, who in his letters to another mathematician, Richard Dedekind complained on his inability to
overcome the contradiction between and the well-known "truth" that a part is smaller than the whole and the
conclusion he reached on the grounds of his new theory, according to which a subset of an infinite set may be
"as big as" the whole set (Cavaill`es, 1962).
9
Of course, some cases of conflicting narratives may stem from differing opinions rather than from discursive
conflict. Discursive conflict should be suspected only in those cases when the conflicting narratives are
factual and the possibility of an error in their construction has been eliminated. A narrative counts as factual if
there are well-defined meta-discursive rules for their substantiation.
Negatives Sfard & Avigail43
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Table 1: Comparison of concepts: communicational conflict versus cognitive conflict
Cognitive conflict Communicational conflict
Ontology
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10
Only too often, communicational conflicts are mistaken for factual disagreements. Most well known
incompatibilities between scientific theories may, in fact, be resulting from communicational conflicts rather
than from correct versus incorrect factual beliefs. Thus, for example, what appears as a straightforward
contradiction between Aristotle and Newton – the former thinker's claim that a constant force applied to a
body results in the body's constant movement, versus Newton's second law of dynamics asserting that
constant force results in a constant acceleration – may, in fact be the outcome of the two men's differing uses
of the word force.
Negatives Sfard & Avigail44
How is it
resolved?
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11
The notion of reasonable communicational agreement can be seen as a communicational counterpart and
elaboration of Brousseau’s idea of didactic contract, that is, of “the system of [students’ and teachers’]
reciprocal obligations” (Brusseau, 1997, p. 31). We do not claim that communicational agreement is sufficient
for the success in overcoming the communicational conflict – we only say it is crucially important for
learning. This is a theoretical assertion, analytically derived from basic tenets of our approach, but our
findings in the present study seem to corroborate it.
Negatives Sfard & Avigail46
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12
The term discourse-for-oneself is close to Vygotsky’s idea of speech-for-oneself, introduced to denote a
stage in the development of children’s language (see e.g. Vygotsky 1987, p.71). These ideas also brings to
mind the Bakhtinian distinction between authoritative discourse, a discourse that “binds us, quite
independently of any power it might have to persuade us internally”; and internally persuasive discourse, one
that is “tightly woven with ‘one’s own world.’ (Bakhtin, 1981, pp. 110-111.)
13
As Vygotsky (1978) reminds us, a sociocultural vision of learning (and, in particular, his own notion zone
of proximal development) must result in “reevaluation of the role of imitation in learning” (p. 87)
14
Let me add that we often insist on children’s own inventions not only because of the learning opportunities
they create, but also because we believe that in this way we show more respect to the learner and, while doing
so, help her or him to be a better person. However, belief in the possibility of children’s unmediated learning
from the world and the wish to sustain the democratic sprit of the classroom discourse are not an indissoluble
whole, and abandoning the former does not necessitate compromising the latter. It seems that realistic
communicational agreement can be cultivated in schools without any harm to the democratic spirit of
classrooms interactions. Indeed, making sense of other people's discourse is not any less creative or
demanding than "reading the codes" of nature.
Negatives Sfard & Avigail48
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Negatives Sfard & Avigail51
- CitationsCitations106
- ReferencesReferences57
- "The special version of discursive conceptualization to be used in my analyses, called commognitive 1 , equates mathematics with discourse, that is, posits that mathematics is a certain well defined form of communication (Sfard, 2007Sfard, , 2008Sfard, , 2012). According to commognitivism, therefore, to think mathematically means communicating – with others or with oneself – in the special way called " mathematical " . "
[Show abstract] [Hide abstract] ABSTRACT: In this chapter, while analysing the event from Chapter 2 and trying to identify the opportunities for learning the teacher has been creating for his students, I am guided by the conceptualisation of learning called commognitive. According to this approach, mathematics is a form of discourse and learning mathematics means becoming capable of participating in this discourse. Explorative and ritualized mathematizing are two kinds of discursive activity, differing in their perceived goals and in the rules of participation. While taking a close look at the teacher's talk, I am asking into which of these two types of mathematizing Mr. T was inviting the learners. My later scrutiny of the two follow up interviews with students is guided by the question of how the learners utilized those opportunities.- "Classroom talk is in this conception considered the most essential cultural tool mediating learning (Lehesvuori, 2013). More recently Sfard (2007 Sfard ( , 2008) uses the term commognition e coined as a blend of communication and cognition e in order to emphasise the indivisibility of these two phenomena. She recommends viewing learning not as acquisition of knowledge, but as participation in a certain discourse. "
[Show abstract] [Hide abstract] ABSTRACT: This study deals with the impact of a teacher development programme focused on the implementation of dialogic teaching practice. Four indicators of dialogic teaching were measured: student talk with reasoning, teachers' open questions of high cognitive demand, teacher uptake, and open discussion. An analysis of video recordings made before and after the programme showed a change in classroom discourse and an increase in the amount of student talk with reasoning, attributed to changes in teacher communication behaviour. The participants were eight Czech teachers in lower secondary schools who took part in a one-year action research teacher development programme.- "Per cultural–historical psychology (Vygotsky, 1930Vygotsky, /1978 ), we conceptualized the interval as a self-generated artifact. Wedged between the subject and the object (Sfard, 2007 ), the interval served as an emergent auxiliary construction mediating the students' participation in the social enactment of a novel cultural practice. The respective work of Saxe (2002) and Radford (2005) clarified the transitional function of cultural forms in the MIT-P activity. "
[Show abstract] [Hide abstract] ABSTRACT: Whereas emerging technologies, such as touchscreen tablets, are bringing sensorimotor interaction back into mathematics learning activities, existing educational theory is not geared to inform or analyze such learning. In particular, educational researchers investigating instructional interactions still need intellectual and methodological frameworks for conceptualizing, designing, facilitating, and analyzing how students’ immersive hands-on dynamical experiences become formulated within semiotic registers typical of mathematical discourse. We present case studies of tutor–student behaviors in a technologically enabled embodied-interaction learning environment, the Mathematical Imagery Trainer for Proportion. Drawing on ecological dynamics—a blend of dynamical-systems theory and ecological psychology—we examine how students develop effective sensorimotor schemes as solutions to interaction problems. We argue that in attempting to enact a new complex skill, learners may improve their performance by spontaneously constructing an attentional anchor—a real or imagined object, area, or other aspect or behavior of the perceptual manifold that facilitates motor-action coordination. We further argue that symbolic artifacts introduced into the arena may both mediate new affordances for students’ motor-action control and shift their discourse into explicit mathematical re-visualization of the environment. We thus implicate symbolic artifacts as ontological hybrids evolving from things you act with to things you think with. Students engaged in embodied-interaction learning activities are first attracted to symbolic artifacts as prehensible environmental features optimizing their grip on the world, yet in the course of enacting the improved control routines, the artifacts become frames of reference for establishing and articulating quantitative systems known as mathematical reasoning.
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