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Learning and teaching mathematics requires the activation of a variety of resources^ which can be grouped around what I consider to be two chief sources of mathematics knowledge: the body and its activity with artifacts, and the activity with signs'. The cognitive significance of the body has become one of the major topics in current psychological and cognitive studies. In particular, the ‘‘Embodied Mind” paradigm (Lakoff & Nunez, 2000) has situated the origins of all human knowledge—mathematics included—in bodily experiences and perceptions. On the other hand, knowledge formation is embedded in cultural and social contexts, and the use of signs becomes crucial in cognition. Such a polarity results in being of particular interest in the context of the leaching and learning of mathematics, a discipline traditionally considered “abstract” on the one hand, and extensively based on perceivable signs on the other.

Many studies in mathematics education research have framed teaching and learning activity within semiotic perspectives, focusing on written semiotic systems such as algebraic symbolism. Only recently has attention been dedicated to considering bodily means of expression as semiotic resources in the learning process, and to looking at their relationship to written mathematical symbolism (see for instance Radford, Bardini & Sabena, 2006 on the role of rhythm in the context of algebraic generalization).

In this chapter, I focus on the contribution of gestures to the mathematics teaching-learning processes in the classroom context. The role of gestures will be outlined by situating them as semiotic resources and looking at how they are closely intertwined with more traditionally studied semiotic systems (such as language and written signs).

The chapter is divided into three parts: theoretical reference points, an analysis of examples from the classroom context, and a final discussion based on the analysis provided.

THE STUDY OF GESTURES

When people talk, they gesture. Gesturing is a widespread phenomenon: it is found in all cultures that have been observed, occurring across a wide range of tasks and ages, even when the listener is not physically present, or cannot be seen. Gestures are part of what is called “nonverbal communication”, which includes a wide- ranging array of behaviors such as the distance between people in conversation, eye contact, voice prosody, body posture, and so on. According to the traditional view of communication, all these acts, although important in framing a conversation,

L /?tidford. G. Scbubring. and F. Sevger feds.). Semiotics in Mathematics Education: Epistemohgv.

History. Classroom, and Culture. 19-3S.

C 2008 Sense Publishers. AU rights reserved.

have little to do with the conversation itself, which is instantiated in uttered speech (“■verbal communication”). In this perspective, gestures are interpreted as nothing more than hand waving, embellishments, means for letting out excess energy, bids for the listener’s attention, or regulators in the communicative exchange. Such a dichotomist view has been widely challenged in the last three decades, starting from the work of Kendon (1980), who was among the first to propose that at least one form of nonverbal behavior—gesturing—cannot be separated from the conversation itself².

Nowadays, research in a number of disciplines (such as psychology and all its branches, cognitive linguistics, and anthropology) is increasingly showing the fundamental importance of gestures not only in communication, but also in cognition. Curiously, it has been the interest in cognition prompted by Chomsky’s view of linguistics as a kind of purely “mental science” that has led to the vigorous investigation of gestures by those interested in language:

If language is a cognitive activity, and if, as is clear, gestural expression is intimately involved in acts of spoken linguistic expression, then it seems reasonable to look closely at gesture for the light it may throw on this cognitive activity (Kendon, 2000, p. 49).

What is “Gesture"?

“Gesture”’ is used in various ways in ordinary language and also in literature. It sometimes refers to any movement that people perform while talking, including movements of the hands and arms, adjustments to posture, self-adaptors (e.g.

touching one’s hair), functional actions (such as picking up something), nervous tics, etc. It is clear that not all such body movements play a part in learning processes;

therefore, it is reasonable to look for definitions and operative distinctions.

In Kendon’s research, gesture is assigned a general meaning, which he organizes in a continuum ranging from spontaneous gestures (which he calls “gesticulation”) at one edge, to sign languages at the other edge (Kendon, 1988):

gesticulation —» pantomimes -a emblems —» sign languages.

At the center of the continuum, we find pantomimes that are namely significant gestures without speech (like twirling a finger around in a circle to describe a vortex without using words), and emblems, which are endowed with a culturally specific standard form and are used in an intentional way to communicate precise meanings (e.g.. the OK gesture). McNeill (2000) analyses this distinction with respect to the relationship to linguistic properties and to conventions, and the character of semiosis. Gesticulation and pantomime⁴ are described as being unconventional and not seeming to obey any system constraints, and therefore lacking all linguistic properties. Sign language, on the contrary, is itself a fully developed linguistic system and is purely conventional. Emblems are in between, showing evident conventionality hut not forming a linguistic system⁵. McNeill concludes that “nothing about the visual-manual modality per se is incompatible with the presence of linguistic properties” (McNeill, 2000, p, 4).

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ON THE SEMIOTICS OF GESTURES

Analyzing conversational settings, McNeill (1992) defines gestures as "‘the movements of the hands and arms that we see when people talk'¹ (McNeill, 1992, p. 1). In this approach, widely adopted in successive research studies in psychology, gestures end up being viewed as distinct but inherently linked with speech utterances⁶. Indeed, in the context of mathematical activities in the classroom, where the focus is not communication perse but knowledge formation, phenomena accounted for as gestures may occur in a wider range of actions beyond mere conversation, such as handling tools or "simply” thinking in silence. A working definition suitable to mathematics learning settings can therefore include in the term “gesture” all those movements of hands and arms that subjects (students and teachers) perform during their mathematical activities and which are not a significative part of any other action¹ (ie. writing, using a tool, ...).

This definition has to be intended in an inclusive way, and is certainly not clear- cut. In any case, whatever definition is taken, it so happens that the notion of gesture remains quite fuzzy, in the sense that there is no hard-and-fast line between what is “gesture” and what is not. Hence, how can one identify gestures and distinguish them from other body movements? Gestures are usually characterized as follows (Kendon, 1996; McNeill, 1992): they begin from a position of rest, move away from this position, and then return to rest. The central part of the movement, generally recognized as expressing the conveyed meaning, is called stroke or peak" it is preceded by a preparation phase (hand/arm moving from its resting place, and usually to the front away from the speaker), and symmetrically succeeded by a retraction phase (hand/arm back to the quiescence). Speakers of European languages usually perform gestures in a limited space in the frontal plane of the body, called gesture space, which goes roughly from the waist to the eyes, and includes the space between the shoulders. However, differences have been detected according to age (with children the space is larger) and different cultural settings⁵¹.

Types of Gestures

Even narrowing the meaning of“gesture” a la McNeill, the term comes to include a variety of behaviors that do not form a single category. In McNeill's study, the following basic gesture types are identified⁹:

- iconic gestures: bear a relation of resemblance to the semantic content of discourse (object or event);

- metaphoric gestures: similar to iconic gestures, but with the pictorial content presenting an abstract idea that has no physical form;

- deictic gestures: indicate objects, events, or locations in the concrete world;

- beats: the hands move along with the rhythmical pulsation of speech, lending a temporal or emphatic structure to communication;

- cohesives: tie together thematically related but temporally separated parts of the discourse.

Iconic, metaphoric and deictic gestures have a main imagistic and pictorial component, whereas beats and cohesives are kinds of discourse-structuring gestures.

Each type uses the gesture space in a different way—i.e. the iconics are mostly in the center part, the deictics more in peripheral regions—and each category has a different story of development in children. However, a same gesture can show features belonging to more than one category. This follows from the fact that the taxonomy is not based on the mere shape or kinesthetic characteristics of the gesture, but on its relationships with external elements, such as the accompanying speech and the task at hand. Gesture interpretation is therefore context-dependent and needs information about the action schemes provided by our experience with objects in the world (Cassel, 1998). Iconic gestures, also referred to as representational (Kendon, 1988; Kita, 2000), through their imagistic features provide information not only on some object that is being represented, but also on the particular point of view the speaker is taking towards it. Metaphoric gestures provide an image of something invisible, an image of an abstraction. For instance, the idea of a topic may be presented as a bounded container supported by the hands: a certain portion of space comes to be used to present something else that is inherently nonspatial.

Deictic gestures are pointing movements and are prototipically performed with the index finger. They have the function of focusing joint attention on a shared reference and are the most context-dependent, since they derive their interpretation essentially from the context in which the listener-speaker interaction takes place¹⁰. In pointing, the movement may not be simply linear, but can follow various patterns, which can have semantic implications and iconic features. Apparently simple, pointing is on the contrary a complex, typically human act. A special case of pointing is called abstract pointing, when there is no actual physical pointed object, rather the pointed empty space houses an introduced reference, for instance an abstract concept about where the speaker has been at a certain moment he is talking about. McNeill recognizes a certain meaning to such pointed space: ‘‘the speaker appears to be pointing at empty space, but in fact the space is not empty; it is full of conceptual significance. Such abstract deixis implies a metaphoric use of space in which concepts are given spatial forms” (McNeill, 1992, p. 173).

As it emerges in the discussion, the taxonomy is not to be intended in a rigid way, rather it is meant to suggest some of the various functional dimensions gestures can contribute to communication. In the next paragraph, we shall see that the strict link recognized between language and gesture has implications that go beyond communication and involve cognition itself.

Gestures, Speech and Thought

in psychological studies, gestures and speech are generally considered as comple­

mentary sides of the same coin: linguistic systems are described as conveying meaning in a segmented, analytic, linear, and hierarchically structured way; gestures, on the contrary, are characterized as global, synthetic, multidimensional and never hierarchical (McNeill, 1992, p. 19). In his semiology, Saussure (1916/1967) attributes the linear-segmented character of spoken language to its un[dimensionality, contrasted to the mullidimensionalsty of meanings. This characterization is taken up by Gold in-Meadow, who contrasts language and gesture features:

ON THE SEMIOTICS OF GESTURES

Language can oniy vary along the single dimension of time. [...] This restriction forces language to break meaning complexes into segments and to reconstruct multidimensional meanings by combining the segments in time.

But gesture is not similarly restricted. Gestures are free to vary on dimensions of space, time, form, trajectory, and so on, and can present meaning complexes without undergoing secmentation or linearization (Goldin-Meadow. 2003.

pp. 24-25).

Language is made of elementary' constituents and its rules of production involve both atomic (single) and molecular (compound) signs. On the contrary, gestures are described as endowed with holistic features that cannot be split into atomic components. The modes of production and transformation are often idiosyncratic¹¹ to the subject who produces them (McNeill, 1992, p. 19 ff). Furthermore, one gesture can combine many meanings (synthetic character) and two gestures produced together do not combine to form a larger, more complex, hierarchically higher gesture¹², as language, in contrast, does: gesture is free to vary' on dimensions of space, time, form, trajectory and can present meaning complexes without undergoing segmentation or linearization, nor being constrained to precise standards of forms (McNeill, 1992).

Being very different forms of expression, each with its own specificity, gesture and speech appear, however, to be intimately related: they are semantically and pragmatically co-expressive, they are essentially synchronous^1-5 in time and meaning, and they develop together in children. From cognitivist perspectives, the benefit gesture brings is often interpreted in terms of lightening the cognitive burden, in an analogue way that writing a problem down can reduce the effort needed to solve the problem. Research findings seem, in fact, to show that gesturing confers a significant cognitive benefit on verbal and visuospatial memory', by reducing the load of working memory and thus allowing it to organize the cognitive resources to solve a given task better (Goldin-Meadow, Nusbaum, Kelly

& Wagner, 2001). Gesture can therefore constitute an embodied form for off­

loading cognitive work into the environment (Wilson, 2002). Concerning the gesture-speech relationship, McNeill puts it in a clear way: "‘[Gestures] are tightly intertwined with spoken language in time, meaning, and function; so closely linked are they that we should regard the gesture and the spoken utterance as different sides of a single underlying mental process” (McNeill, 1992, p. I).

This quotation, coming from the very' beginning of McNeill’s cornerstone book, highlights an important dimension in gesture studies in psychology, namely their role in cognitive processes. Though differing in the specific positions they assume on the issue, the most recent studies in cognitive psychology and linguistics agree in recognizing the importance of the function of gestures in cognition, up to upholding that gestures and language should be viewed within a unified conceptual framework (Goldln-Meadow, 2000; Kita, 2000; McNeill, i992)^u.

In McNeill’s discussion, gesture is related to the important dialectics of individual and social planes:

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The gesture supplies the idiosyncratic, the personal, and the context-specific aspects of thought, to be combined with the socially regulated aspects that come from the conventions of language. Such a combination implies a dialectic of gesture and language in which the gesture provides the momentary context of speaking and language carries this individuality to the social plane where it is categorized, segmented, reformatted, and dressed up for the world. Putting these themes together, we can conceive of thought as fundamentally an inner discourse in which gestures play an intrinsic part (McNeill, 1992, p. 2).

Going in the direction of extending Vygotsky’s claim that “thought is not merely expressed in words; it comes into existence with them" (Vygotsky, 1934/1986, p. 218), McNeill (1992) interprets the essential speech-gesture cognitive unit as supporting an active constitutive role of gestures on thought: “gestures do not just reflect thought but have an impact on thought. Gestures, together with language, help constitute thoughr (p. 245, emphasis in the original).

In other words, according to a Vygotskian perspective, we can frame gesture and language in a unifying semiotic mediating¹⁵ function: gestures together with language provide the materiality in which thought can arise and unfold.

TAKING PEIRCE’S SEMIOTIC APPRO ACM

To analyze the role of gestures as semiotic resources in the classroom, they have to be inserted into a semiotic approach allowing a very general definition of “signs".

This is well accomplished in Peirce’s pragmatic theory.

Peirce defines a sign as a triad composed by the sign or representamen (that which represents), the object (that which is represented), and the interpretant:

A sign, or representamen, is something which stands to somebody for something in some respect or capacity. It addresses somebody, that is, creates in the mind of that person an equivalent sign or perhaps a more developed sign. That sign which it creates I call the interpretant of the first sign. The sign stands for something, its object. It stands for that object, not in all respects, but in reference to a sort of idea, which I have sometimes called the ground of the representamen (C.P., 2.228).

The object is the referent, that which the sign represents. Before being interpreted, the representamen is pure potentiality. A sign can “stand to somebody for something in some respect or capacity” only because the relation “to stand for” is mediated by an interpretant. The interpretant is not the interpreter of the sign;

rather it is what guarantees the validity of the sign even in the absence of the interpreter. The interpretant is another representation referred to the same object. It can be an equivalent significant—but Peirce would speak of “sign”—in another semiotic system (be. a drawing to explain a word meaning), an index to the single object, implying an element of universal quantification (“all the objects like this”), another definition in the same semiotic system (i.e. salt for sodium chloride), an emotive association (i.e. dog for fidelity), the use of synonyms. The list does not

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ON THE SEMIOTICS OE GESTURES

have the effect of reducing the interpretant to the intensional properties of a content and hence to the series of denotations and connotations of an expression: for Peirce, an interpretant can even be a complex discourse, inferential])' developing all the logical possibilities implicated in the sign; an interpretant can even be a syllogism deduced from a regular premise. Furthermore, it can be a behavioral response, a disposition and many other things as well. What is striking and innovative in Peirce’s introduction of the interpretant in the very definition of sign is that the sign is thereby endowed with an intrinsically dynamic character. In fact, the interpretant is a sign that translates and explains the previous one, and this other sign in its turn requires another sign as interpretant, and so on in a chain of infinite interpretation, establishing a process of dynamic unlimited semiosis. Peirce’s idea of semiosis involves the relation Sign-Object-lnierpretant as inreducibly triadic. He often insists that the triadic relation is genuine, in the sense that it cannot be reduced to any of the couples formed by its terms.

The notion of sign introduced by Peirce is very general: every phenomenon, simple or complex as it may be, can be interpreted as a sign entering into a semiotic process¹⁰. The essential feature for something to be a sign is to be able to represent something else to somebody, where “to represent” means “to stand for, that is, to be in such a relation to another that for certain purposes it is treated by some mind as if it were that other” (C.P,, 2,273). Adopting a Peircean semiotic perspective, everything entering into a semiotic process is a sign, and therefore a larger variety of phenomena, included body movements and gestures, can be considered as semiotic resources.

Besides the general characterization of signs, a relevant aspect of Peirce’s theoiy with respect to the present research is its pragmatic character. In fact, it considers the context of sign production and reception and defines the sign through its action on the interpretant. This is a main difference from Saussure’s program for linguistics, where there is no provision for the study of the actual contexts in which

Besides the general characterization of signs, a relevant aspect of Peirce’s theoiy with respect to the present research is its pragmatic character. In fact, it considers the context of sign production and reception and defines the sign through its action on the interpretant. This is a main difference from Saussure’s program for linguistics, where there is no provision for the study of the actual contexts in which