Gesture in collaborative mathematics problem-solving

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Abstract

In this paper we examine the significance of gestures in a setting where two students try to make sense of, and solve, mathematical problems involving speed and time. We are particularly interested in exploring the claims that gesture serves various signaling functions in collaborative problem-solving communication generally and in mathematics problem-solving more specifically, and that gesture has a diagnostic role for the collaborators and for teachers. The overall purpose of our paper is to illustrate the integral role of gesture in dyadic communication where core problem domain concepts may be difficult to explicate.

Introduction

  • Lucy:

    A bouncing ball. I think a bouncing ball would be like D.

  • Fran:

    Or E.

  • Lucy:

    D. No, it wouldn’t. Course it wouldn’t. Stupid. Wouldn’t be like E, it’s not, a ball’s not gonna, like 

  • Fran:

    Go backwards. It’s going to go woo-woo, woo-woo. It’s not going to do that. It’s going to go slcchh! It’s gonna go speed up and slow down, it gets to the top and then.

  • Lucy:

    It’s gonna go like this, it’s gonna go like this.

  • Fran:

    I reckon it would go like that. It would be like B again.

  • Lucy:

    Like B? It’s not like B.

  • Fran:

    D.

  • Lucy:

    Yeah, that’s what I reckon. It’ll go bo-ing except it’ll bounce a little lower each time. I reckon. So I reckon it’ll be like D.

  • Fran:

    Yeah. Mm. I don’t know.

A reader encountering this dialogue will likely be at a loss to understand what the participants in the interaction are talking about. A person listening to an audiotape of this dialogue would be better placed to interpret the interaction: prosodic cues such as stress, intonation, rate of speech, and pauses would all assist in comprehension. Information about the context of the interaction would obviously help — the dialogue is an extract of a problem-solving episode from a mathematics lesson concerning graphical representations of speed as a function of time. Even with the assistance of prosody and context, however, the students’ dialogue remains obscure to the outsider, verging on incomprehensible. How, then, can the students themselves arrive at some form of effective, mutually accepted understanding of and solution to the problems before them? We suggest that the students’ gestures have an integral and multifaceted role in achieving collaborative understanding (Reeve & Reynolds, 2001). Further, we suggest that the gestures made by collaborating students provide a rich source of diagnostically useful information for teachers. Specifically, we propose that gesture has at least three functions in dyadic communication. First, gesture is used to achieve, maintain and refocus joint-attention on the problem to be solved. Second, gesture reinforces and extends the meaning conveyed by the students’ speech, particularly where the language of the problem-solving domain is unfamiliar. Third, it is possible that in some circumstances, gesture may index a state of cognitive uncertainty that may herald the onset of changes in understanding.

A hallmark of students working in collaborative learning settings is the co-construction of meaning. Particularly where there is limited access to the formal language specific to the problem domain, students must try to dynamically co-construct an understanding of problems in order to solve them. The understanding so achieved may be inadequate (Brown, Campione, Reeve, & Ferrara, 1992; Siegler, 1996); nevertheless, how such co-construction occurs is a dominant theme in the study of learning and development (Brown and Reeve, 1987). The overarching question is, how do participants identify the nature and extent of the other’s understanding? A closely related question is whether participants are aware of changes in their own and each other’s understanding. An important issue, therefore, is whether it is possible to identify transitions that signal participants’ changes in understanding. Of course, such transitions are difficult to identify and are likely to have varying etiologies involving such matters as domain knowledge, interest, engagement, metacognitive abilities and motivation, all of which may well co-vary during the collaborative process.

The likely link between gesture and verbal language has long been recognized (Efron, 1941; Werner & Kaplan, 1963; Wundt, 1921/1973), with mounting evidence that gesture is an important precursor of spoken language (Iverson and Goldin-Meadow, 1997, Iverson et al., 2000). During the pre-linguistic period, gesture seems to have at least two specifically communicative functions: first, as a way of achieving and maintaining joint-attention (Morissette, Ricard, & Decarie, 1995) through deictic pointing and touching (Masataka, 1995; Tomasello & Camaioni, 1997; see also Rome-Flanders & Cronk, 1995), and second, as a means of symbolic communication (e.g., cupping the hand to indicate the desire for a drink) (Acredolo and Goodwyn, 1997, Goodwyn and Acredolo, 1993).

There is a growing body of evidence that gesture is implicated in various aspects of cognition even after the acquisition and mastery of spoken language. Studies of children’s early mathematical understanding have highlighted the importance of gesture as children learn to count (Alibali & DiRusso, 1999; Graham, 1999), as they develop an understanding of the number sequence (Steffe, 1988; Steffe & Cobb, 1988), as they discover composite units (Steffe, 1992, Steffe, 1994), and as they begin to understand fractions (Steffe & Olive, 1996). The way that gesture operates in such situations is not clear, although it has been speculated that gesture may act as an external representation that facilitates the mapping from concrete to symbolic referent (Graham, 1999). Nevertheless, it is recognized that gesture is a valuable source of information about what it is that children themselves understand about these aspects of mathematical cognition (Steffe, 1992). Recent research also suggests that, in some forms of mathematical problem-solving, gesture may indicate impending cognitive change. In studies of children’s solutions and explanations of number conservation and mathematical equivalence problems, Goldin-Meadow and her colleagues observed that children who were acquiring a novel concept sometimes gestured when explaining their understanding of that concept to the experimenter (Alibali and Goldin-Meadow, 1993, Church and Goldin-Meadow, 1986, Goldin-Meadow and Alibali, 1995, Goldin-Meadow et al., 1993, Goldin-Meadow et al., 1993). Interestingly, children’s spoken explanations were often wrong while their gestures indicated a different, sometimes more complete, conceptual understanding (gesture–speech mismatch). Goldin-Meadow and colleagues suggest that such children were experiencing cognitive dissonance, a state in which several ways of solving a problem are simultaneously activated (Goldin-Meadow et al., 1993, Goldin-Meadow et al., 1993), and that such children are more likely to benefit from instruction in the particular concept. Goldin-Meadow and colleagues claim that cognitive dissonance as evidenced by gesture–speech mismatch is an index of impending cognitive change.

The possibility of gesture being implicated in the development of children’s mathematical understanding is intriguing. The question then becomes whether the gestures made by children when learning mathematical concepts, particularly gestures that may indicate the onset of cognitive change, have any impact upon other people. The effect of gesture–speech mismatch on recipients has recently been explored in a series of studies in which adults (both teachers and non-teachers) (Alibali et al., 1997, Goldin-Meadow et al., 1992; see also Perry, Woolley, & Ifcher, 1995) and children (Church et al., 2000, Kelly and Church, 1997, Kelly and Church, 1998) viewed video vignettes of children solving problems and exhibiting both gesture–speech match and gesture–speech mismatch. Goldin-Meadow and colleagues claim that the results of these studies demonstrate that the occurrence of gesture–speech mismatch affects the way that both adults (whether or not they have teaching experience) and children interpret children’s conceptual understanding. However, these studies did not involve face-to-face situations; thus, there was no opportunity to explore whether the dynamics of collaborative interaction would result in different findings.

The findings of Goldin-Meadow and her colleagues have obvious implications for the co-construction of meaning in collaborative learning settings. Indeed, the significance of gesture in classroom learning and other settings is being increasingly recognized by various researchers (Crowder, 1996; Crowder & Newman, 1993; Moschkovich, 1996, Roschelle, 1992; Roschelle & Clancey, 1992), who now regard gesture as having significance in its own right in collaborative problem-solving discourse.

In Roschelle’s studies, pairs of students were videotaped while engaged in discovery learning about velocity and acceleration in a computer microworld. Roschelle notes that gestures were essential to interpreting students’ speech. Specifically, Roschelle (1992) proposes that gestures are “metaphors-in-action,” with the dual function of coordinating the participants’ joint-attention and of constructing a shared understanding of the meaning of the concepts of velocity and acceleration.

In a detailed examination of the relationship between middle and high school students’ gestures and talk during science lessons, Roth (2000) found that students who lack sufficient domain language to talk about scientific phenomena typically use gestures that correctly identify and explain those phenomena. He observes that, as scientific talk emerges, gestures occur before spoken utterances, but that speech and gesture co-occur once students attain familiarity with the domain under consideration and the appropriate domain language.

While this research constitutes a useful analysis of the role of gesture in collaborative and classroom learning situations, several important theoretical and methodological issues remain. First, the function of gesture in cognitive transition is still not clear, although more recently there has been speculation that gesture may play a causative role in cognitive change: first, by externalizing aspects of thought and thus increasing working memory capacity, and second, by enabling students to represent to themselves ideas that they cannot yet put into words (Goldin-Meadow, 2000; Goldin-Meadow & Alibali, 1995). Second, with the exception of Crowder’s (1996) and Roth’s (2000) analyses, the description of gestures made by students is generally imprecise. Fine-grained description and classification of gestures may allow a better insight into the interweaving of speech and gesture that may further elucidate the nature of collaborative cognitive change. Third, no research has yet examined the understanding that students themselves have of their own and others’ gestures as they occur in naturalistic collaborative problem-solving settings. In the following analysis of two collaborative problem-solving episodes, we address four issues. First, does gesture assist in achieving and maintaining joint-attention in collaborative problem-solving? Second, does gesture convey understanding of a problem domain in the absence of sufficient command of the language of that domain? Third, does the co-occurrence of gesture and prosodic elements such as rate of speech, pauses, and emphases, indicate shifts in cognitive understanding? We are of the view that the association of gesture with changes in speech tempo and pauses has a functional similarity to gesture–speech mismatch. Finally, what understanding do participants in the collaborative discourse have of their own and others’ gestures?

The two problem-solving episodes that we examine here form part of the database of the Classroom Learning Project (Clarke, 2001). The topic of the mathematics lesson from which the episodes are drawn was the graphical representation of speed as a rate. The teacher began the lesson with a led discussion of graph interpretation tasks in the mathematics booklet produced by the school (see Fig. 1).

Next the teacher discussed a student’s bus trip to school, drew a diagram of the journey, and constructed a speed–time graph on the blackboard. The teacher then led a class discussion interpreting this graph, pointing out different aspects of the motion depicted, including constant acceleration from zero velocity, constant velocity, and deceleration to zero velocity. Students were then instructed to work in pairs. The first task was to assign appropriate speed–time graphs to different scenarios such as a car slowing down and reversing, the speed of a ball thrown into the air at different points of its journey, a stone dropped from a great height, and a bouncing ball. The second task was to examine a speed–time graph of a school bus on its morning run (see Fig. 2) and to describe the bus trip from that graph.

In our analysis, we examine protocols of two brief exchanges between two students, Fran and Lucy. The protocols include the students’ speech and gestures. Gestures appear on the same line or lines as the accompanying speech. As we are interested in the possible interaction between the occurrence of gesture and prosodic elements such as speech tempo and pauses, we have italicized speech that was slower than normal, while pauses longer than 1 s appear in brackets at the appropriate place in the protocol.

Section snippets

Episode 1: discussion of speed–time graph for a bouncing ball

The first problem-solving episode we analyzed involved deciding which of the graphs in Fig. 1 best represented a bouncing ball. The students’ protocol appears in Table 1.

We found two examples of gestures being used to maintain joint-attention on the problem: at turn 1.5 Lucy taps the page to refocus the dyad’s attention on her initial choice of graph D following the earlier discussion about graph E, while at turn 1.6 Fran immediately shifts the attentional focus by pointing at her own choice of

Episode 2: discussion of graph representing the journey of a school bus

The second episode involved the description of a bus trip based on inspection of the speed–time graph in Fig. 2. The relevant protocol appears in Table 2.

In this episode, gesture plays a major part in establishing and maintaining joint-attention. Lucy uses gesture frequently to direct joint-attention to the problem. At turn 2.7 she taps the page, stressing the importance of looking at the graph. At turn 2.9 she points to the section of the graph showing the bus acceleration. At turn 2.13 she

Conclusions and suggestions for future research

The last decade has seen an almost exponential growth of the gesture literature. The dominant areas of research have concerned the function of gesture in the prelexical period of child development, and the likely implication of gesture, particularly where it conflicts with speech, as a marker of and possible participant in the cognitive development of older children (and, indeed, adults). Research into the role of gesture in collaborative learning, too, is beginning to expand.

In this paper we

Acknowledgements

Our thanks to David Clarke for the opportunity to review tapes and transcripts, to the other participants in the Classroom Learning Project for their comments on our work, and to Les Steffe for his most helpful comments on an earlier version of this article that was presented at the annual meeting of the American Educational Research Association, New Orleans, April 2000.

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