Gesture and the Communication of Mathematical Ontologies in Classrooms

■ Matthew WolfgramWisconsin Center for Education ResearchUniversity of [email protected]

Gesture and the Communication ofMathematical Ontologies in Classrooms

This article is an analysis of a high school geometry teacher’s gesture and embodied socialaction in the context of classroom instruction, and its role in the communication of an implicittheory of the ontological status of mathematical knowledge. In particular, I argue that theteacher’s bodily performance in the classroom setting communicates the Platonic position ofabstract-realism—that mathematical entities are ontologically real abstractions—which isassociated with the high epistemological authority and social efficacy of mathematics. Thus, inaddition to the teacher’s use of language to reference mathematical knowledge, I document herpatterns of embodied social action, which communicate an additional and ideologically specifictheory about the ontological status of the items thus referenced. I propose the concept of ametaphor of participation—an iconic projection based on the pattern of bodily participationin the speech event—as a tool to theorize the bodily communication of ideologies. [gesture,language and body, STEM, classroom discourse, mathematics]

. . . all things are numerable.John Stuart Mill, System of Logic (1970[1884]:II.4.7)

This article focuses on the role of bodily communication—hand gestures but alsoother features of bodily performance—in the communication of ontologies of math-ematical knowledge in the context of U.S. public schooling. I argue that the coordi-nation of gesture and other features of embodied social action form an iconic sign bywhich teachers communicate ideological content about the ontological status of math-ematical entities. I call this type of ideological sign a metaphor of participation and it isa central goal of this article to develop and demonstrate the effectiveness of thisconcept for the analysis of social action. This theory is supported by the empiricalclaim of this article, which is that while the philosophy of mathematical knowledge isnot the focus of explicit linguistic reference in the classrooms I observed, teachersemploy a combination of gesture and bodily signs to communicate ideologicalcontent regarding the ontological status of mathematical knowledge. This article willpresent a detailed analysis of one teacher’s classroom instruction, which illustrateshow bodily performance can communicate such ideologically specific content as aphilosophy of mathematics.

The evidence for this embodied communicative process comes from my analysis ofa large corpus of video recordings of teacher–student interaction, which I collectedwith colleagues in the learning sciences (including educational and developmentalpsychologists), as part of an effort to research how teachers’ gestures might enhancemathematics education (Alibali et al. 2014; Nathan et al. 2013). Along with ethno-graphic observations and interviews, video recordings of teacher–student interaction

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Journal of Linguistic Anthropology, Vol. 24, Issue 2, pp. 216–237, ISSN 1055-1360, EISSN 1548-1395. © 2014by the American Anthropological Association. All rights reserved. DOI: 10.1111/jola.12049.

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were collected in a variety of public school mathematics classrooms, including pre-algebra, algebra, and geometry, as well as high school engineering/technology class-rooms (equaling a total of 56 dual camera recording sessions in 14 separateclassrooms). As an illustrative case of the role of gesture in the communication ofmathematical knowledge, I will provide below a detailed analysis of the classroompractice of one public high school geometry teacher. It is important to note that theembodied communicative process that I detail in this article is common throughoutthe large corpus of observations and video recordings of the math classrooms. Inparticular, the process is frequent in the setting of didactic communication, in whichthe teacher stands at the front of the class, the students are seated in their desks facingforward, and mathematical inscriptions are written on the whiteboard, smart board,or transparency projector. There are other participation formats in these classrooms,such as student individual seat work and peer group desk work, or computer labwork, which lack this didactic goal of the presentation of lesson content—and theytypically also lack the embodied communicative process that I will describe. Thus, thephenomena of the use of gesture and bodily signs to communicate mathematicalontologies is a common practice in the classrooms I observed but it is tied to the goalsof didactic teaching in the traditional classroom format, where new or review lessonmaterial is written on the board and discussed.

The teacher presented in this article exemplifies the embodied communicativepattern; in fact, it was during the analysis of videos of her classroom when I firstnoticed this communicative process, which I later identified across the wider corpus.The two transcripts presented below are iterations of the behavior which are briefenough to present in their entirety in this article and which also demonstrate all therelevant and interesting communications features related to the argument.

The case will demonstrate that while ideologies of mathematical knowledge arenot the explicit focus of the teacher’s pedagogy or linguistic reference, the teachercommunicates through gesture a particular set of assumptions about the ontologicalreality of the mathematical entities. This ontology is central to the ideological author-ity and social efficacy of mathematics. I suggest that a close analysis of the role ofgesture in the communication of ideologically specific content is needed to advancethe sociopolitical analysis of language beyond the historically persistent Cartesiandualism opposing language on the one side and body on the other.

There is a history of the anthropology of visual and nonverbal communicationwhich has worked to transcend this dualism, starting with the pioneering work ofGregory Bateson and Margaret Mead (1942), Edward T. Hall (1963), and RaymondBirdwhistell (1970), and later developed by Adam Kendon’s (1994, 2004) anthropol-ogy of gesture, and continued with research involving micro-interactional analyses ofembodied multi-partied interaction by Charles Goodwin (2006) and Marjorie HarnessGoodwin (2006). The goal of the present article is to further this tradition of theanalysis of bodily communication by connecting it to an explicit theory of ideologicalcommunication. That is, I take the position that it is not only that bodies communicate,but that it is also through such a process of bodily communication that a social realityis produced in significant ways.

In the next section, I discuss the issues related to the ontology of mathematicalknowledge and the connection between these ontologies and ideologies of theauthority, efficacy, and rationality of mathematics.

Mathematical Ontology and Ideology

The role of mathematical knowledge in organizing reality is central to the social andcognitive organization of experience in post-industrial capitalist economies. The late-19th-century utilitarian philosopher John Stuart Mill’s grand statement in theepigraph of this article, that “all things are numerable” (1970[1884]:II.4.7), whileempirically incorrect, was prophetic of the current situation of modern and rational-ized institutions (Harris 2005). Good government, good society, good economy, good

Gesture and the Communication of Mathematical Ontologies in Classrooms 217

health, and good knowledge—all things dear to the utilitarian cause—have in factbecome subjected to the rationalizing power of mathematics. As Ian Hacking (1990)has demonstrated, since Mill’s time in 19th-century Europe, reality has increasinglybeen ordered through a process of mathematical rationalization, including, he argues,the phenomenon of chance (unpredictable events), which was brought to disciplineby the development of a new probabilistic mathematics (a.k.a. statistics). The expan-sion of mathematical rationality was integral to the expansion of European power andeconomic domination in the context of European colonization, as for example, in thecase of the introduction of numeracy practices in 19th-century British India to facilitatecolonial governance by imposing a quantitative ordering of the subject Indian popu-lation (Appadurai 1993). Thus, there is a special and rational kind of power engen-dered by the deployment of mathematics to order experience by rendering itcalculable.

The naturalization of the authority of mathematics to represent reality imparts aveneer of objectivity to interventions into the economy and society (“the numbersdon’t lie”), interventions which are themselves political-economic in nature (Godelier1973). The education of this “trust in numbers” (Porter 1995) is a central accomplish-ment of science, technology, engineering, and mathematics (STEM) education beyondthe teaching of mathematics skills. Thus, the two major goals of STEM education—thetraining of STEM professionals and the generalized education of a “mathematicallyliterate” citizenry (e.g., Steen 2001)—have the embedded ideological goal of produc-ing people with this faith in numbers.

In spite of the significant role played by mathematics in our late-capitalist society, thenature of mathematical knowledge and its relationship to reality is far from clear orself-evident. In fact, there is an ongoing debate that has spanned the 20th centuryamong professional philosophers and mathematicians on such fundamental questionsas the ontological status of mathematical entities and whether or not mathematicalstatements are true (Benacerraf and Putnam 1983; Shapiro 2005). The dominant posi-tion in the history of Western philosophy, called Platonism, holds that mathematicalentities are “abstract forms.” The argument of Platonism is that entities such asnumbers, theorems, and so on, are real but that they do not exist in space and time.Aristotle apparently held this position (Lear 1982; Mignucci 1987). This claim ofontological abstract realism is important to consider because it relates to what philoso-phers call epistemology (and I call “ideology”), which is the assessment of the truthstatus of mathematical knowledge: if mathematical entities are real then mathematicalstatements can be true. Thus, Platonism is both a theory about the nature of mathemati-cal reality, and an ideology about the authority of mathematical representations of thatreality. As such, Platonism is also an ideological statement of the authority of math-ematicians to make such truth claims—and to have those claims accepted as truthful.Platonism is an excellent example of what I have elsewhere called an “ideology oftruth” (Wolfgram 2010)—a cultural/ideological theory about what kinds of proposi-tions or truth claims can be accepted at truthful and authoritative.

The mathematician Paul Bernays (1985[1935]) has suggested that Platonism is thestandard theory of practicing mathematicians. Mathematicians act as if their objectsare Platonic, and it is only when queried by philosophers that critiques of Platonismare acknowledged or that the possibility of non-Platonic foundations of mathematicsare entertained (but not often seriously). Another mathematician, Timothy Gowers(2002:17) explains the Platonic disposition among his colleagues, that “. . . therecertainly are philosophers who take seriously the question of whether numbers exist,and this distinguishes them from mathematicians, who either find it obvious thatnumbers exist or do not understand the question” (on the Platonic foundations ofmathematics also see Davis and Hersh 1981). There are several other positions aboutthe ontological foundations of mathematics which have been developed by philoso-phers and philosophically minded mathematicians—all with their own pros andcons—although none of these other philosophies has attained anything like theunmarked and intuitive statues of the Platonic ideology.

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Some of these other interpretations are non-Platonic. Formalism, for example,developed by the famous mathematician David Hilbert, reduces mathematics to themathematical forms themselves (Detlefsen 1986; Zach 2006), and Logicalism reducesmathematics to the structure of logic (Frege 1980[1884]; Whitehead and Russell1910[1962]). Both of these approaches seem to sidestep the troubling issue of ontologyby reducing math to a presumably universal domain, the forms themselves on the oneside and the structure of logic on the other. The only non-Platonic position to under-mine (or trivialize) mathematical truth value is Fictionalism (Field 1989), which is theradical denial of the ontological reality of mathematical entities. This claim has attimes been identified with post-modern (or social constructivist) interpretations ofmathematics. Following this position, mathematics can be both useful and interest-ing, but it is ultimately also highly trivial because mathematical entities are notontologically real, and thus, mathematical statements cannot be true. Most positions,however, are more sophisticated reinterpretations of the Platonic ideal, which alwaysultimately argue that mathematical entities are real and as a consequence, that thestatements of mathematicians are true. Nominalism, for example, posits that theabstract forms of mathematics are real because they correspond to empirical objects,and Structuralism (like structuralism in linguistics and anthropology) is the positionthat mathematical entities are themselves not real but that the structure of theirrelationship is so considered real. Naturalism in mathematics is another refinement ofPlatonic abstract realism. The Naturalist argument is that because mathematical rep-resentations are indispensable to the accurate modeling of scientific representations ofreality, and because scientific knowledge is knowledge of an ontological reality, thenmathematical entities can be said to be ontologically real and mathematical statementscan be said to be true (Putnam 1972; Quine 1969).

There is a subtle mysticism that is implicit in the Platonic position that mathematicsis real yet it does not exist in space and time. Formalist and Logicalist reductions, andsophisticated reinterpretations of Platonism by brilliant contemporary philosophers,never completely succeed in rationalizing this mystical quality of mathematical enti-ties. These philosophies are motivated by the desire to affirm the rightness of the truthstatus of mathematical statements (and thus, avoid Fictionalist and postmodernistextremism). The cognitive scientists George Lakoff and Rafael Núñez (2000) havedelineated some of the fundamental assumptions of this mysticism, which they havetermed the “the romance of mathematics”: that mathematics is abstract yet real; thatit is part of the structure of reality yet it is also a transcendental feature of the universeitself (thus, it is somehow both empirical and transcendent); and that it is the struc-ture of logic, and thus all intelligent beings in the universe and intelligent computerswill reason mathematically. So, the mathematical rationalization of the world, whichmakes experience reducible to calculable quanta, is itself based on the subtle mysti-cism of the ontological reality of nonsensible objects. This ontological assumptionmust be socialized as part of our acceptance of the rationalization of experience inpostindustrial capitalist society.

There has been some preliminary research on the potential role of teaching thephilosophy of science along with science content in STEM classrooms (Monk andOsborne 1997). If there is robust empirical evidence that the incorporation of philoso-phy aids in the goals of science education, then there will be reason to considerexpanding the program to math education as well. However, the current state ofaffairs is that math education at the secondary level is conducted in the United Stateswith little or no explicit consideration of this interesting philosophical debate on theontology of mathematical entities (although classes on Mathematical Foundations orthe Philosophy of Mathematics are sometimes offered at the university level indepartments of mathematics and philosophy, respectively).

There is, however, a philosophy of mathematics that is communicated by teachers’gestures. Mathematical entities are represented as both objective forms, with visual-spatial and material qualities, and at the same time, as abstract and generalizedideations. This representation of mathematics as both objective and abstract follows


the basic ontological claim of Platonism. As we have seen, this is the default interpre-tation of practicing mathematicians and it is a serious area of philosophical inquiry.This position of abstract realism seems counterintuitive, and yet the objectification ofabstractions is a common representational process in scientific communities of prac-tice, for example the use of data visualizations like charts, graphs, and so on, tovisually objectify concepts and processes (Latour 1986). I argue in this article that thisontology accomplishes a fundamentally ideological goal, which is to confer an aura ofauthority to the truth claims of the people who deploy mathematical representations.

It is important to make clear that this ideology of mathematics is not an explicit orconscious focus of classroom instruction. If fact, the teachers that I interviewed tendedto characterize their student learning goals in very pragmatic terms. They hoped toteach their students logical thinking and argumentation, effective communicationand collaboration skills, math skills that might be helpful in adult life and, above allelse, to impart math credentials and skills that would help them gain entrance and tosucceeded in college. Thus, if teachers have a philosophy of mathematics at all, itwould be most appropriate to characterize it as a form of pragmatism. The point ofthis article is that a very different and specific ideology of mathematics is communi-cated through the teachers’ bodily practices.

In the next two sections, before describing the classroom context and the transcriptdata, I first develop a critique of the role of the body in social theory and then Ipropose an explicitly semiotic mechanism of embodied ideological communication.

The Embodiment of Ideology

There is a tendency in social theory to assume a disembodied conception of ideology.The goal of this section is to critique that assumption and to establish the need insocial theory for a specifically semiotic theory of embodied ideological communica-tion. For example, while ideology theorist Terry Eagleton argues that social control ismainly accomplished through a process of bodily disciplining (1991, followingFoucault 1977), he represents ideology as pure cognition dedicated to explaining therightness of that discipline. Eagleton argues that the criterion that distinguishes anideology from a belief is the presence of political motivations on the part of theproducers and/or the interpreters. This recourse to the sociopolitical context of theinterpreters is where embodiment enters in despite Eagleton’s stance, since interpret-ers are of course differently embodied and positioned within a mode of production,and thus, possess differing political motivations.

As Judith Irvine has rightly observed (1989), the opposition of language (as pureideation) and speech (as social and material), which is so central to the structuralistapproach to language, is a reinscription of the mind/body (ideal/material) dualismthat pervades the history of Western philosophy more generally. In a similar repro-duction of the fundamental Cartesian dualism, of which Marx himself was so critical,neo-Marxist ideological theory has reinscribed the opposition of mind and body in anovel form, as an opposition between language and ideology on the one hand, andbody on the other. And so, within the logic of language/body dualism, languagecommunicates the ideology and the docile body is conditioned to accept the contentthus communicated.

Pierre Bourdieu’s “theory of practice” (1977) is the most comprehensive andsophisticated social theory that is explicitly based on a bodily mechanism of social-ization and communication. Bourdieu argues for the important role of bodily habits(objectified and totalized as “the habitus”) in the communication of implicit socialvalues and assumptions about reality, articulated in the following statement:

. . . nothing seems more ineffable, more incommunicable, more inimitable, and, therefore,more precious, than the values given body, made body by the transubstantiation achieved bythe hidden persuasion of an implicit pedagogy, capable of instilling a whole cosmology, an


ethic, a metaphysic, a political philosophy, through injunctions as insignificant as “stand upstraight” or “don’t hold your knife in your left hand.” (1977:94)

In Bourdieu’s work, however, the mechanism of the habituation of social values—bodily practice—is ill-defined and undertheorized. That is, Bourdieu’s mechanism ofhabitus explains that a body can be socialized to accept ideologically specific propo-sitions, such as cosmologies and so on, but it is not clear how bodily practices couldthemselves communicate such ideological content. Furthermore, it is clear fromresearch on language socialization, that such ideological socialization involves thecoordination of both bodily and linguistic patterns of interaction (for example,Burdelski 2010). What is required, I suggest, is a semiotic theory of bodily communi-cation which articulates how bodily action (in coordination with language) can itselfbe a ground for ideological signs.

Thus, in the next section, I will articulate a bodily mechanism of ideologicalcommunication. I propose and provide evidence for the usefulness of the semioticconcept of a metaphor of participation, which is an iconic ideological projection basedupon the pattern of bodily participation in the context of social action. The concept ofparticipation (first introduced by Erving Goffman 1979) involves the social andsequential coordination of talk between participants, which has become a key unit ofanalysis in social interaction research, specifically because the concept highlights thesocial, spatial, bodily aspects of action. Such patterns of action are sequences of bodilyactivity, coordinated between participants in a particular semiotically constitutedvisual-spatial-material ecology, and involving the participants’ movement within,attention toward, and manipulation of this ecology (C. Goodwin 1981; M. H.Goodwin 2000).

Importantly, metaphors of participation are based on a very similar semioticprocess as the one described by Judith Irvine and Susan Gal (2000) for languageideologies, involving a projection of an iconic relationship between language andsome other level of sociocultural organization (“fractal recursivity”), and involving aconsequent “erasure” of sociolinguistic diversity. A metaphor of participation is thatsame process, except that the ground from which recursion is projected is not lan-guage but bodily action and the target domain includes language but also othersociocultural domains as well. I take the position that while metaphors of participa-tion are not ideologies of language per se but rather ideologies of bodily action andcommunication, body-based metaphors of participation often coordinate with lan-guage ideologies. I also suggest that in some cases language ideologies may originateas metaphors of participation.

The approach I am developing is consonant with the trend toward a broadapproach to sociolinguistic style (Eckert and Rickford 2001; Irvine 2001), which incor-porates not only linguistic features but also dress, ornamentation, gait, and otherbodily symbols and dispositions. Furthermore, an embodied approach to languageideological production advances a unified material-semiotic perspective central to apolitical economy of language. The foregrounding of body in the process of ideologi-cal production also advances the language ideological literature by highlighting as yetundertheorized embodied social practices as mechanisms of language ideologicalproduction. I argue in this article that part of that process involves the semioticmechanism of metaphors of participation, which I further develop in the next sectionin terms of its fundamental character as a body-based iconic projection of socialaction.

Metaphors of Participation as a Body-Based Iconic Projection

I will argue that key cultural ideologies and assumptions about the nature of realityare communicated by body-based metaphorical projections. A metaphor of partici-pation is a sign that postulates an iconicity between this pattern of action and theideology of the speech event. The metaphor establishes an iconic correspondence


from the pattern of embodied action to some other sociocultural domain, includingknowledge, social relations, and language. Such icons are diagrammatic in nature inthat the behavior forms a pattern that itself becomes the ground of the sign(Mannheim 1999).

Social action is a compost of constituent semiotic resources and potentials that arenegotiated and deployed in an embodied social activity context (Goodwin 2000).There are three such constituent features which require local management to form thebasis of embodied metaphorical signs. Social action that is iterative forms a patternwhich is the first semiotic condition of a metaphor of participation. Iteration plays akey role in Charles Sanders Peirce (1955) semiotic theory, in particular, in the phe-nomenology that underlies the theory. An isolated case of social action is an instanceof what Peirce called the condition of Secondness—an experience, an event, amoment of self/other contrast. It is the second and subsequent instances of suchsimilar experience that realizes a pattern between events over time, which is whatPeirce identified as the state of Thirdness. Social actions that are ritualistically iterativematerialize an embodied pattern or rhythm that can be the basis of an iconic projec-tion. For an iterative social action to become such a sign, however, there are two otherconditions that are subject to local management between participants in the socialsetting.

Features of the iterative pattern of social action must be regimented to be bothcontiguous and distinctive. Diagrammatic iconicity is constituted by this regimentationof units to form a pattern of separate yet related features. Iteration, contiguity, anddistinctiveness are features of social action which form the semiotic potential of socialaction to itself be interpreted as an ideological sign. These features are managedlocally in the context of social interaction—which constitutes the performance of ametaphor of participation. Social action, I suggest, always has this potential for dia-grammatic iconicity. This is a compelling type of ideological sign process because itsmaterial base is social action itself, which conveys a natural and self-evident qualityto the metaphorical projection.

George Lakoff (Lakoff 1993; Lakoff and Johnson 1980) has argued that metaphor isfundamental to human thought and that such thinking is often based upon thepsychomotor experience of bodily action. The argument makes intuitive sense whenapplied to online activity sequences, where activity is environment-focused andsocially distributed. However, even forms of thinking which in the Western intellec-tual tradition are typically considered highly rational or abstract cannot be viewed asfundamentally off-line or disembodied. As reasoned and sedentary a discipline asmathematics (Lakoff and Núñez 2000), and even the mathematics of phenomena thathumans cannot experience, such as that of infinity (Núñez 2005), has been demon-strated to be at least partially based on metaphorical projections of common bodilyexperiences. Lakoff and colleagues—as philosophers and cognitive scientists—address the metaphorical processes of an idealized human body, with typical phe-nomenologically intuited experiences. A radical implication of their research foranthropologists, however, is to highlight the potential cognitive significance of highlyspecific bodily patterns of social action, which provide the basis for specific ideologi-cal projections. Such metaphorical projections are not based upon an idealization ofbodily experience, but rather—as illustrated by Victor Turner’s (1974) work on meta-phorical projections of ritual structure—they are based upon actual experiences ofbodily activity which occur in a performative context. It is the specificity of the bodilyecology in the context of performance that is the material basis of metaphors ofparticipation.

Gesturing in Classroom Ecologies

We will now consider the classroom ecology and the teacher’s communication anduse of gesture in a high school geometry classroom. The term “classroom ecology” inthe education and learning science literature refers to the spatial relations between


participants in the classroom, including the seating arrangements and the presence ofinstructional tools (MacAulay 1990). I also use the term to refer to physical arrange-ments of bodies, tools, furniture, and other accoutrements in the space of the class-room. An additional and important feature of my use of the concept is that Iemphasize the non-neutral quality of the classroom ecology. The physical setting isnot a stage for the performance of teacher—student interaction, but rather, it is aconstitutive feature of the interaction itself. It is the movement of the teacher’s bodywithin this particular classroom ecology which is the basis of a metaphor for thecommunication of mathematical knowledge.

The transcripts that we will consider below are based on a video-recording of anadvanced-level geometry classroom. The school is a public high school in a suburbansection of a midsize city in the Midwest of the United States. The classroom and theschool has a diverse student body, both in terms of ethnicity and socioeconomicstatus, due in large part to careful designing of the school catchment zones on the partof the district administration to encourage such diversity. There is a major researchuniversity in the city as well as a technical college. As advanced geometry students,the students in this classroom have successfully managed prior educational gate-keeping courses for the math curriculum, such as Algebra and Advanced Algebra,and the teacher’s sense of her students was that the majority of them will not onlygraduate high school but go on to some form of postsecondary education. The highschool honors track is a consequence of the schedule of math classes needed forcollege entrance exams and applications. Students who finish basic algebra in middleschool are able finish the full slate of college preparatory math courses by graduation.Students who elect to take a project-based Connected Math course—which is some-times thought of as a remedial math course by teachers and parents—tend to start thesequence of courses a year behind the honors track students, and thus, they areunable to finish the full sequence (i.e., pass calculus). Thus, the difference between theregular and the honors classes is not primarily a difference in lesson content butrather that the regular students are a year behind the honors students in the mathcurriculum.

When I approached Mrs. Green about observing and video-recording her class-room she immediately recommended her honors class over her regular geometryclass because, in her own view, the better discipline of the honors students allowedher to experiment more with her pedagogy. This was a very common belief of themath teachers who I observed, although I personally never found the non-honorsclassrooms especially unruly.

The geometry teachers at the school had elected to follow a hybrid curriculum,employing a traditional geometry textbook, but also incorporating projects and prob-lems from a project-based geometry curriculum that had recently been purchased bythe district. This project-based pedagogy contrasts with the acquisition of mathemati-cal skills through many iterations of problem-solving practice, sometimes referred toas the “plug and chug” method. At the same time, the teachers found that theirstudents lacked needed basic algebra skills to succeed at geometry, and the traditionaltextbooks involve more algebra practice than the project-based curriculum. So, theteachers at the school employ both the traditional “transmission” metaphor of teach-ing and learning with respect to algebra content, as well as the metaphor recentlyincorporated into the district through math education reform, that learning is aprocess of discovery, and that the method of discovery should be as practical andhands-on as possible.

Both metaphors of learning are incorporated into the habitual rhythm of Mrs.Green’s honors classroom. Each curricular unit spanned several days. The studentscycled between the lecture, discussion, and assessment of geometry content in theirclassroom, and then move over to the computer lab across the hall. There, the stu-dents took partners and experimented by manipulating the properties of digitalgeometric representations on the computer, ideally, discovering the mathematics forthemselves. Afterward, the students formed peer groups to discuss their work.


After the lab, the teacher and her students reconvened in the classroom toreview the material through lecture/discussion, which continued into the introduc-tion of the next lesson. The examples presented below are taken from the moredidactic lecture/discussion portion of the class, when mathematical content istransmitted to the students and the experiences gained from the lab and otherclassroom activities are discussed as a class. Elsewhere, colleagues and I have inves-tigated student gesturing in math classrooms, for example, in the context of peergroup discussions and lab activities (Nathan et al. 2012). The focus here is on theteacher’s gesturing because it is the institutional role of the teacher that isforegrounded as “on stage” during the classroom didactic instruction. It is in thiscontext that the teacher claims interactional authority and presents the curriculumas the cannon of knowledge to which the students will be held accountable onsubsequent assessments.

The classroom ecology of teacher-led instruction will be familiar to readerseducated in U.S. schools. Mrs. Green stands at the front of the classroom in frontof a whiteboard, where there are inscribed geometric representations. She alternatesher attentional focus between the board and the students, who sit individually intheir assigned desks, which are oriented toward the front of the class. The class-room is structured as a panopticon (Foucault 1977), so that the teacher canobserve that the students are attending to the lesson, with the appropriate booksopen to the correct pages, and all other required tools visibly present and at theready. This spatial organization highlights the power asymmetries between theteacher and the students through its regulation of social relations in terms of thephysical arrangement of the desks. As indicated above, however, the notion of thepanopticon still assumes the language/body dichotomy, with the students’ bodiesmade ready to receive instruction and the teacher’s body positioned to perform thatinstruction. The content, i.e., the knowledge, is separate from these bodies and iscommunicated through language to the properly oriented bodies of the students.However, as I will show below, this perspective remains silent on how thebody can itself serve as a channel of communication for ideologically specificcontent.

Gesture in Math Education

Gesture and bodily communication is particularly rich and complex in the context ofmathematics classrooms. This is in part because of the semiotic complexity of therepresentation of mathematics, which includes the use of linguistic representationsbut also of complex visual inscriptions such as drawings, as well as a system ofmathematical symbols (O’Halloran 2000). With respect to the role of gesture in com-municating math content, educational research has shown that teachers use gesturesto ground instruction (Nathan 2008), construct and interpret math representations(Roth 2003), scaffold student learning (Alibali and Nathan 2007; Valenzeno et al.,2003), link representations together (Alibali et al. 2014), communicate additionalinformation about math concepts not included in the talk (Goldin-Meadow et al.1999), and to visually embody difficult mathematical concepts (Alibali and Nathan2012). It has also been argued that mathematical thinking is fundamentally bodybased (Lakoff and Rafael Núñez 2000), and there is empirical evidence for thisphilosophical position from research on math learning in classrooms (Nemirovskyand Ferrara 2009; Roth 2009). It is noteworthy that such research on gesture andembodiment in math education settings is supported by empirically rich descriptionsof gesture and bodily action.

In this analysis, I focus on the combination of gestures and other features of bodilycommunication in teacher performances that function not just to convey content/curriculum, but also as ideological signs.

There are several kinds of gestures and gesture-like actions that are employedin mathematics classrooms. These gestures can be classified into two general


categories, which I call contextualizing and textualizing gestures. It is the use of thesetwo types of gestures which imbues classroom space with semiotic value, and it isthis spatialization of value and the movement between these classroom spaceswhich constitutes the classroom metaphor of participation. The categories ofcontextualizing and textualizing were designed to analyze the discursive productionof ritual authority (Hanks 1984; Kuipers 1990). For example, in the context ofWeyyewa ritual oratory on the Indonesian Island of Sumba, some ritual genressuch as divination gain their ritual efficacy and authority from the physical settingand the social relations among the mutually coordinated participants. A diviner candiagnose the ancestral origin of an unfortunate event by sacrificing a chicken andsplaying the entrails to be interpreted by the concerned parties. The languageof divination contains a high frequency of contextualizing discursive features,such as locatives which index the spatial context of the speech event (including theparticulars of the chicken entrails), stand-alone pronouns which index thesocial relations of the speech event, and discourse markers which index theinternal deliberative state of the speaker. These features underscore the basis of theauthority of the ritual utterances to act on the particular social context of the speechevent, where responsibility for the negative sanctioning by the ancestorsmust be diagnosed and an appropriate form of action must be charted. That courseof action will invariably involve other types of ritual speech such as blessings, pla-cation songs, ancestral narratives, and other poetic genres, addressed to theaffronted ancestors and performed by a professional ritual specialist. Thesetextualizing genres are characterized by the delivery of the poetic text organized intopaired couplets—a.k.a. “the words of the ancestors.” Textualizing genres have alow frequency of discourse features that index the context of the speech event,such as the locatives, pronouns, and mental state discourse markers common indivination. In contrast, Kuipers reports a low frequency of unpaired words (i.e.,more couplet speech) and also a high frequency of intertextual discourse markerssuch as “so that,” which connect couplets into a unified text. The authority pro-duced by contextualizing discourse is based on the particular characteristics of thespeech event, what Peirce (1955) refers to as a “token,” and the authority oftextualizing discourse is based on the generalization of a rule (in Peirce’s terms, a“type”). This type/token semiotic distinction is similarly produced by thecontextualizing and textualizing effects of the gestures employed in the mathemat-ics classroom.

Contextualizing gestures are oriented to mathematical inscriptions on thewhiteboard and they emphasize the particularities of the representation.Contextualizing gestures are indexical—or at least partially indexical—in that theconnection to the context is required to understand the meaning of the gesturalcommunication. Such gestures include (1) pointing to or touching inscriptions andtheir parts, (2) tracing parts of the inscription with a pointed hand or with a pen, (3)iconic gestures laminated over the inscription, and (4) the writing and drawing witha pen of additional inscriptions. Such contextualizing gestures particularizethe focus of attention on the specific properties of the manifested instance ofmathematical knowledge—i.e., the inscription on the board. The teacher’s gazeshifts between an orientation toward the students, who are seated facing theboard, and the board itself, where she constructs and references the mathematicalrepresentations.

Textualizing gestures are quite different. Typically, they occur in the space locatedbetween the board and the students. Such textualizing gestures include (1) “conduitmetaphorical gestures” that represent the math concept as an object with visual-spatial and material qualities (McNeill 1992), (2) iconic gestures of the mathematicalrepresentation which are decoupled from the whiteboard as a representational space,and (3) pointing to text inscriptions (e.g., terminology written in text-books) whichdenote the decontextualized and abstracted concepts exemplified by the mathemati-cal representations on the board. Textualizing gestures such as these represent


the mathematical knowledge as a thing in itself, detached from the instance ofmathematical knowledge represented on the board. The teacher’s talk reinforces thecontrast between contextualizing gestures—which co-occur with demonstratives andother indexical forms—and textualizing gestures—which co-occur with abstractnoun forms.

The teacher’s gesturing, and patterns of attention and action, constitute the domi-nant classroom metaphor of participation involved in the process of the communi-cation of the Platonic ideology of mathematics. There are three constituentideological processes realized by this metaphor of participation. I call the first par-ticularization, which is the production and direction of attentional focus on math-ematical inscriptions as particular instances or examples of mathematicalknowledge. Particularization is the production of tokens of knowledge. The secondideological process is abstraction, which is the opposition of particular instances (ortokens) of mathematical representation and the concepts (or types) that they exem-plify, and the representation of those concepts as abstractions. The construction ofa type/token opposition is fundamental to the process of abstraction and theboundary between instances and abstractions must be clearly regimented. A par-ticular circle inscribed on the white board is a token, for example; and a theoremthat describes the characteristics of circles in general is a type. The oppositioncreates the appearance that mathematical entities are pure abstractions—evidencedby particular instances of mathematical representation. The third ideological processis reification, which is the representation of that abstraction as an object form. Theprocess is common in STEM classrooms, where linguistic processes such asnominalization and practices of inscription represent the characteristics of experi-mental events (processes, actions, and relations) as objects (Keane 2008; Massoudand Kuipers 2008; Radford 2013). Taken together, the processes of particularization,abstraction, and reification produce the special object of mathematical activity, theabstract form. It is important to note that reification does not erase the abstractnature of the mathematical entity, but rather reification introduces a fundamentalduality into the representation of mathematical knowledge. Mathematical entitiesare abstractions, as well as things. That contradiction is their very nature—at leastaccording to the dominant ideology.

We have now defined the concept of a metaphor of participation as a diagrammaticiconic projection from the pattern of bodily action to another sociocultural domainassociated with the event, including the domain of language but also of knowledge,social relations, and so on. The analysis of such metaphors of participation require avisual-spatial description of the ecology of participation, and a language for describ-ing the types of action—in this case, contextualizing and textualizing gestures—employed to embody the metaphor within the particular classroom ecology. Theinstantiation of this metaphor by the teacher’s communicative practice in the class-room realizes three constituent ideological processes (particularization, abstraction,and reification), which in total, constitutes the special focus of mathematics education,the ontologically real abstract form of Platonic mathematics. This article now turns tothe task of documenting a specific embodied metaphor of participation in a math-ematics classroom.

The Transcript Evidence

I will present two transcripts of Mrs. Green’s geometry classroom. Note in thetranscript below that gesture and bodily orientation are highly significant. Nearlyall the lines of text are accompanied by a screen-capture, as well as an italicizeddescription of the bodily communication. The camcorder is positioned at the back ofthe class, shooting over the heads of the students who are seated in their desks,facing the board and the teacher (a second camcorder captured the students). InLines 1–8, the teacher is mainly oriented toward the inscription on the board. Thebodily communication is what I have characterized as contextualizing gesture. On


Lines 9–10 you will notice the teacher’s fairly dramatic shift in bodily orientationaway from the whiteboard and toward the student addressees. During this shift onLines 9–10, and continuing on Line 11, the teacher’s bodily communication is whatI have characterized as textualizing gesture. Finally, on Line 12, the teacher mediatesbetween the contextualized and textualized classroom spatial ecology, by linkingwith her body, the particular mathematical representation as an instantiation of ageometric theorem, which is an abstract and formalized rule about the nature ofmathematical entities. The bodily construction and movement within this classroomspatial ecology is an example of what I have called a metaphor of participation(Transcript 1).

Transcript 1: “this theorem on this page”

1 T: So .. wherever this inscribed angle ispoints to inscribed angle of the diagram

2 T: it's half of the arcangle icon, followed by tracing of the lines that form the angle of the arc

3 T: that it interceptsdraws “X” variable on the arc

4 S: okay

5 T: so if that arc is Xturns to face the student

6 S: okay

7 T: this is gonna be half of thatdraws “½X” inside the inscribed angle

8 S: okay

9 T: so this theorem on this pagepoints to a student’s book on their desk (underlined segments are beats)

10 T: that you're trying to provewalks toward students seated in the front of the class (no photo)

11 T: is trying to proveiconic conduit metaphor, holds theorem that students are “trying to prove,” (underlined segments are beats)

12 T: wh::y:: … that has to be halfleft hand holds metaphoric gesture, right hand backward thumb point to diagram(underlined segments are beats of the backward thumb point)

Transcript 1“this theorem on this page”


Line 1 starts with the teacher pointing and touching a particular angle which isinscribed in a circle. Mathematical representations are complex and multifeatured,and teachers such as this geometry instructor often assist students in identifying therelevant parts and their relations. Charles Goodwin (1994, 2005) has described howgestures such as pointing can be used to socialize a discipline specific style ofperception, what he calls a “professional vision.” The teacher’s pointing gestures areattempts to socialize the students with just such a disciplinary vision, by “highlight-ing” the figure of the relevant angle and the ground of the circle and other parts. Thesubsequent two lines (#2–3) involve additional highlighting of relevant features of theinscription. So, on Line 2, the teacher forms an iconic gesture of the relevant angle andplaces that iconic hand shape over the inscription. The gesture is both iconic (based onresemblance) and indexical (based on its contiguity with the inscription). This indexi-cal grounding continues as Mrs. Green moves her hands from that angle position,back and forth along the two vertices of the angle. This movement highlights that thevertices intercept the circle and form an arc. Actually, the circle is segmented intothree arcs but the highlighting function of the iconic gesture and tracing motionforegrounds the particular arc opposite the relevant angle. The teacher adds aninscription that codifies the highlighting process by labeling that relevant arc, “X,” onLine 3, after which several students chime “okay” in approximate unison. “So if thatarc is X,” the teacher turns to face the students, pauses to check that the students areattending, and then returns her orientation to the board, “this (the angle) is gonna behalf of that (the arc)” (Line 7). At that moment, she adds the inscription “1⁄2X” insidethe inscribed angle. The teacher has highlighted two features of a complex math-ematical representation and illustrated their relationship, which is then inscribed onthe representation itself in the form of two additional inscriptions.

So far in this transcript we have considered Lines 1–8, where the teacher employscontextualizing gestures, including pointing and touching the inscription, tracing andiconic gestures laminated over parts of the inscription, and writing inscriptions, whichhighlight parts of the drawing and their relationship. The meaning of these gestures isindexically tied to the particulars of the inscription—in fact, it is the act of this indexicalgrounding which determines the inscription as a particular object of discourse. Suchcontextualizing gestures focus attention on the particular features of the inscriptionand, in doing so, they highlight the specificity of the inscription as an instance ofmathematical knowledge. The inscriptional space, i.e., the whiteboard at the front ofthe class, is the stage for this particularization of mathematical knowledge.

There is a dramatic shift in the teacher’s bodily orientation away from this space ofparticularization (Line 9). This shift signals an abstraction, which I argue entails asubtle transformation in the ontological assumptions being communicated about thenature of mathematical knowledge. On Line 9, the teacher turns away from the boardand points four times to a student’s book which lies open upon her desk. This shift inbodily orientation marks a shift toward the general rule or concept that is exemplifiedby the mathematical inscription on the board. This bodily shift decouples the tokenfrom its context. This is what Mrs. Green says: “So this theorem, on this page, thatyou’re trying to prove, is trying to prove why that [angle] has to be half [of theintercepted arc].” The theorem is written in the student’s book. Because of the pan-optic structure of the classroom ecology, such a reference to an inscription in aparticular student’s book can be taken as metonymic of all the students’ books upontheir desks, wherein can be found an inscription of the theorem in question. “So thistheorem on this page” (Line 9)—each underlined segment contains a co-occurringpoint, which beats in time with the rhythm of the talk. Mrs. Green then walks threesteps toward the students in the class as she utters Line 10, “that you are trying toprove.” On Line 11, she forms a conduit metaphor gesture, holding the theorem in herhands and conferring upon it an object-like ontology. This is followed on Line 12 bya dual gesture, where her left hand remains as the conduit metaphor gesture and herright hand does a backward thumb point to the mathematical inscription, saying“why that [angle] has to be half [of the intercepted arc].”


The space of the whiteboard is the ground for inscriptions of mathematical knowl-edge (tokens), and the teacher’s bodily shift toward the students, with their booksupon their desks, is associated with an abstraction of mathematical knowledge (i.e.,type formation). The teacher’s shift between contextualizing and textualizing ges-tures, and back again to contextualizing gestures, is the instantiation of the discourse-ideological processes of particularization (e.g., teacher’s orientation towardinscriptions), abstraction (e.g., teacher’s bodily shift away from the board), andreification (the representation of concepts as object forms). This shift between context-tied and context-decoupled gesturing is paralleled and reinforced by the teacher’s useof language. Contextualizing gesture co-occur with indexical features of languagewhich are tied to the visual context of the inscription on the board. In Transcript 1, forexample, note the demonstratives that indicate the particulars of the visual inscrip-tion, that are underlined: “this inscribed angle” (Line 1) and “this is gonna be half ofthat” (Line 7). Then, on Line 9 the focus of reference and the indexical ground of thedemonstratives becomes the abstract object represented by the noun form “theorem,”in the context, “this theorem on this page.” The demonstrative ground shifts back tothe visual inscription in the end of the transcript (Line 12), “that has to be half.”

We will now consider in more detail the role of textualizing gestures in thereification of mathematical knowledge. An example of reification in Transcript 1 is thebeating point to the inscription of the theorem in the student’s book. Another exampleof reification is the conduit metaphor gesture on Lines 11–12. The teacher actually holdsthe knowledge in her hands and then, still holding the knowledge with one hand, tiesthe knowledge back to the example on the board by using a backward thumb pointwith the other hand. Conduit gestures which represent knowledge as a quantity, thatcan be contained and transported, are common in Euro-American society (and notuniversal: other societies have other ways of spatializing knowledge [McNeill 1992]).

In Transcript 2 below, Mrs. Green is again producing the typical classroom ecology,with examples on the board and students in their desks with their textbooks. In lines1–14 there is a high occurrence of contextualizing gestures oriented toward theinscriptions on the whiteboard. These include pointing to parts of the inscription(Lines 1, 7, 14) tracing (Lines 4, 10), covering gestures (Lines 3–4, 9–10), and drawingtick-mark inscriptions (Lines 8, 12–13). The highlighting and covering gestures iden-tify the relevant parts of the inscription for attentional focus. On Lines 14–15 theteacher and students manage the shift from particular token of mathematical inscrip-tion to a general rule, in this case, the rule of the “transitive property” (Transcript 2).

Transcript 2: “the transitive property”

1 T: so really they're saying all three of these segments are the samediagrams the relationship of the segments

2 T: right?turns to students

3 T: let’s just forget about this one right here for a secondcovers part of drawing with right arm

Transcript 2“the transitive property”


11 S: Mm mm

12 T: So we could say I'm just gonna do a little double tick markdraws double tick mark on PB

13 T: and a double one on thatdraws double tick mark on PC (no photo)

14 T: and so now can we connect this and say that they're all the same? points to PA, PB, and PC

4 T: are these two segments gonna be the same?covers part of drawing with left are, traces two segments

5 S: yes.

6 T: that's what you did in the lab yesterday .. right?

7 T: that's what we just proved right here .. right? walks over to other section of the whiteboard, points to part of inscription

8 T: is that this one's gonna be congruent to this onedraws tick mark on PA draws tick mark on PB

9 T: now if I cover up this circle herecovers opposite side of the drawing with left arm

10 T: is PB gonna be the same as PC?traces PB traces PC

Transcript 2Continued


22 T: so if PA is congruent to PB point to PA and then to PB

23 T: and PB is congruent to PCpoint to PA and then to PB

24 T: then PA is also congruent to PC point to PA and then to PC

25 T: okay

15 S: transitive propertyteacher forms metaphorical gesture for concept

16 T: by what property?lowers right hand of prior gesture and extends right hand to assign turn to student

17 S: the transitive.

18 T: the transitive property

19 T: so it's just like you know if A equals B and B equals Cwalks toward board, iconic gesture of transitive propertyand then turns toward connecting A, B, & Cthe students

20 T: then A equals Ccontinuing iconic gesture of transitive property, connecting A & C

21 T: we’re just connecting all of themcontinuing iconic gesture,connecting parts A, B, & C

Transcript 2Continued


On Line 14, Mrs. Green illustrates through language and through a series ofpoints that if PA and PB are equal, and if PB and PC are equal, then it can beconcluded that PA and PC are also equal. This is an exemplification of the rule inalgebra termed “the transitive property,” here applied to the context of geometry.Pointing to the lines PA, PB, and PC in sequence (Line 14), the teacher asks, “. . . canwe connect this and say that they’re all the same?,” after which the teacher turnstoward a student requesting a turn and forms a conduit gesture, as if to hold theconcept in her hands and offer it to the student. The student takes the cue on Line15 by identifying the concept of the “transitive property” previously exemplified onthe white board. On Line 16, the teacher asks the student to repeat his answer, “bywhat property?,” and physically hands the concept to the student. The teacher andthe student echo the answer, “the transitive property,” on the subsequent Lines17–18. The shift from specific case to abstract concept accomplished in this interac-tion involves (1) a shift away from the white board as the indexically groundedspace of particularization and (2) the use of a conduit metaphorical gesture to rep-resent the concept as an object form, with visual-spatial and material qualities. Wesee that that shift in orientation and the gesture itself accomplish the reification ofthe mathematical knowledge, which is produced as an object and referenced assuch by the participants.

The process of textualization continues on Lines 19–21. The teacher remains in theapproximate space where she performed the conduit gesture, in between the whiteboard and the students, now facing the students as a group. On these three lines sheperforms with gesture an icon of the transitive property. This icon is decoupled fromthe space of particularization—the white board—and represented as an object form,materialized by the teacher’s hands and located in space between the teacher and herstudents. The event concludes on Lines 22–25, where, similar to the conclusionof Transcript 1, the teacher links the abstract object produced and realized in theclassroom to the particular instance of mathematical knowledge inscribed on theboard. In this case, the teacher shifts her orientation back toward the board, andperforms again an icon of the transitive property, in effect, re-laminating the abstractobject produced in the classroom over the inscription of mathematical knowledge.

Again, as was the case in Transcript 1, the contrast between contextualizing andtextualizing gestures is paralleled by the teacher’s talk, with ample evidence ofindexical talk “all of these segments” (Line 1), “this one right here” (Line 3), “thesetwo segments,” and so on until Lines 15 and 16 when the student and the teacherused the abstract noun form “transitive property.” This shift between the use ofindexical forms and abstract noun forms reinforces the contrast betweencontextualizing and textualizing gestures, and the ideological entailments that theyproduce.

Conclusions: Language and Body

This article has documented the role of gesture, body, and the spatial ecology ofclassroom interaction in the particularization, abstraction, and reification of math-ematical knowledge. The relationship between language and body in this setting isone in which gesture and bodily signs communicate an ideological theory of theontology of mathematical knowledge. We considered transcript evidence of thisprocess from a geometry classroom, focusing in particular on examples where theteacher communicates or establishes a connection between a specific case (i.e., atoken) and a general rule, principle, or abstract concept (i.e., a type). The teacher’stoken-identifying communication is indexically grounded by the body, dominated bybodily and attentional orientation to inscriptions, and reinforced as a particular occur-rence by writing and drawing and by copious hand pointing and tracing gestures.The abstraction of this instance is accomplished by an often dramatic shift in bodilyorientation away from the token’s indexical ground, toward the students who sitfacing the teacher and the board. This bodily shift decouples the token from its


context, and then the teacher employed conduit metaphoric and iconic gestures torepresent the exemplified type, holding the mathematical knowledge in her handsand conferring upon it an object-like ontology.

The ground for this classroom metaphor of participation is this semiotic construc-tion of the classroom space, with the white board produced as the space of particu-larization, and the space between the board and the students produced as the space ofabstraction. It is also the teacher’s movements within this performance space soconstituted. The teacher’s social action iterates between these spaces, which are regi-mented as both distinctive and contiguous, and thus form a diagrammatic iconicpattern. Projected from this ground are the ideological processes of particularization,abstraction, and reification, which produce in the classroom the mathematical entityof the “abstract form” of Platonic mathematical ideology. I submit that this ideologythat mathematical entities are abstract objects that do not exist in space or time is asubtle kind of mysticism, and that the hyper-rationalization of experience through themedium of mathematics in post-industrial society assumes this ontology of math-ematical entities, both for STEM practitioners but also for mathematically literateconsumers. The socialization of this implicit assumption through schooling confersgreat authority on mathematics as a medium of representation.

I have argued that it is the materiality of the body itself and its ecological contextthat causes this representation of mathematical knowledge as possessing this abstractobject-like ontology. The analysis suggests that an analysis of the role of body in theproduction of ideologies requires a specifically semiotic theory of embodied ideo-logical communication. This theory must take as its starting point an integrativeapproach to the relationship of language and body. Such an embodied theory ofideological signification also requires a serious rethinking of the concept of ecologicalcontext. The particular classroom ecology that I have described is not, ultimately,merely a neutral stage for bodily communication—it constitutes the performancein ideologically specific ways. I submit that the concept of a metaphor ofparticipation—as an iconic projection of the pattern of bodily participation—providesa specific semiotic mechanism for embodied ideological communication that does notassume a language/body dualism. The Platonic ideology of mathematical knowledgeis not produced through explicit meta-linguistic reference in the math classrooms thatI have observed. Rather, it originates in the body of the teacher. I propose that thehypothesis of the bodily origin and communication of some ideologies is a potentiallyproductive area for future research. I also propose that in cases where ideologies areproduced through explicit meta-linguistic reference, that if these ideologies are coor-dinated with bodily metaphors of participation, then such ideological forms will likelybe very central, effective, and conservative features of the society.

The project of the sociopolitical analysis of language was initiated in linguisticanthropology by the framing of a political-economic approach to language, whichrequired an explicit and sustained critique of the dualisms of language and speech inlinguistics, and of idealist and materialist ontologies in social theory more generally(Irvine 1989). I suggest that the sociopolitical analysis of language has the potential tofurther expand by the development and application of a semiotic theory of embodiedideological communication—coordinated of course with an equally explicit and sus-tained critique of the comfortable and often hidden opposition of language on the oneside, and body on the other.

Acknowledgments.

I am happy to acknowledge that this research was supported by a generous grantfrom the National Science Foundation titled “Tangibility for the teaching, learning,and communicating of mathematics” (NSF-REESE #DRL-0816406, 2008-2013).Martha Alibali and Mitchell Nathan have provided mentorship, friendship, andsupport of many kinds during this research and I am so grateful. The nascent anthro-pology of STEM education developed in this article is in many ways a response to


their intellectual guidance. I have received detailed and wise criticisms of earlierdrafts of this manuscript from Mark Sicoli, Daniel Ginsberg, JLA Editor AlexandraJaffe, and a particularly insightful anonymous reviewer. Thank you all a great deal.Last, I want to thank my former colleagues at the University of Alabama, Departmentof Anthropology, for their great collegiality over the past few years.

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Gesture and the Communication of Mathematical Ontologies in Classrooms

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