children’s gestures and the embodied knowledge of geometry

MIJUNG KIM, WOLFF-MICHAEL ROTH and JENNIFER THOM

CHILDREN’S GESTURES AND THE EMBODIED KNOWLEDGEOF GEOMETRY

Received: 14 January 2009; Accepted: 9 September 2010

ABSTRACT. There is mounting research evidence that contests the metaphysicalperspective of knowing as mental process detached from the physical world. Yeteducation, especially in its teaching and learning practices, continues to treat knowledgeas something that is necessarily and solely expressed in ideal verbal form. This study ispart of a funded project that investigates the role of the body in knowing and learningmathematics. Based on a 3-week (15 1-h lessons) video study of 1-s grade mathematicsclassroom (N = 24), we identify 4 claims: (a) gestures support children’s thinking andknowing, (b) gestures co-emerge with peers’ gestures in interactive situations, (c) gesturescope with the abstractness of concepts, and (d) children’s bodies exhibit geometricalknowledge. We conclude that children think and learn through their bodies. Our studysuggests to educators that conventional images of knowledge as being static and abstractin nature need to be rethought so that it not only takes into account verbal and writtenlanguages and text but also recognizes the necessary ways in which children’s knowledgeis embodied in and expressed through their bodies.

KEY WORDS: embodiment of learning, geometrical knowledge, gestures

INTRODUCTION

When asked to talk about knowledge, many individuals point to theirheads as if to suggest that this is where knowledge resides. But doesknowledge have to be in the head, encoded as it is in the memory andprocessors of computers? Take the following description from a second-grade mathematics classroom, where the teacher is standing near theblackboard and the children are sitting in their chairs. This is the secondlesson of a unit in which the classroom teacher and one of the researchershas used the word “geometry”. Twelve children can be seen in the videoclip, in which they are reviewing the previous lesson.

The teacher asks the children to predict what the movements of theobjects might be once they are set down the incline plane (i.e. rolling,sliding, and stacking). While the teacher speaks, three children—Ben,Shawn, and Max—gesture along with the teacher’s talk without saying aword. There appears to be neither verbal speech nor intention of directcommunication with others. There is no intended audience for their

International Journal of Science and Mathematics Education (2011) 9: 207Y238# National Science Council, Taiwan (2010)

gestures. The children’s eyes and upper bodies are not oriented towardparticular individuals. It seems they simply gesture by and for themselvesas the teacher’s speech goes along. Their hands are responding, and wemay see how the gestures embody visual aspects of geometric concepts.While listening to the teacher, the students’ bodies, through gesturing,enact and bring forth emergent forms of knowing. It is well-known fromthe literature (see review in Roth, 2001) that such gestures are prevalenteven among blind speakers. We surmise that it is these gesticulationsserve as another possible site where the emergence of the children’sgeometrical knowing can be observed (Table 1). As a means for

TABLE 1

The emergence of children’s gestures

Time Verbal interaction Children’s gestures without speech

00:00 Teacher: …And what were wetalking about yesterday?Cindy, what was one of them?

Ben makes a shape ofsphere with his two hands.

Cindy: circles …(inaudible)00:29Teacher: Ok, so we talked about,wasn’t it a

00:35

circle that we were using or whatwas it?

00:42 Students: a sphere00:43 Teacher: a sphere, because you

know that a circle is flatand a sphereis name for something like a ball.What did a sphere do if it did not slide?

00.58 John: it rolled. Shawn starts rolling hishands before the teacherfinishes her sentence.

01:00 Teacher: And we talked about howrolling means it turns over andover again.

01:05 Teacher: And if it slides, it stayson its bottom and goes down tothe ground. It doesn’t turn over andover again. It can turn a bit but italways stays on its bottom, isn’t itand it goes down.

John turns his right handtoward the left hand.

01:23 Teacher: What is stack? We mightstack chairs.

Max, without looking at theteacher, starts stacking hishands. He repeats the actionseveral times.

MIJUNG KIM, WOLFF-MICHAEL ROTH AND JENNIFER THOM208

expressing understanding not only to/for oneself but also makingavailable to/for others, gestures are constitutive parts of conceptualunderstanding (Roth & Thom, 2009b). Accepting this view of gestures aspersonally and collectively accessible forms of conceptual understanding(Roth & Thom, 2009a) then suggests there might very well be vast areasof thought that are not directly related to speech. There is much ofchildren’s thinking that is not captured when we only consider theirverbal expressions. Before and without speech, children seem to alreadypossess prelinguistic thinking processes; in fact, all linguistic concepts arebetter termed post-kinetic concepts (Sheets-Johnstone, 2009). How thendo children express their thinking if not by verbal manners? And howmight we assess children’s conceptual understandings that are notarticulated in verbal forms?

Our paper contributes to research about the embodiment of mathemat-ical cognition by presenting a study in which we observe and explicatehow the students exhibit conceptual knowing in and through their bodies.It has become a platitude to state that gestures are but one of the ways inwhich human beings engage with others and how they are an integralaspect of communicating and cognition—that is, of commognition (e.g.Sfard, 2007). Drawing on illustrations of children’s gesture-basedinteractions during classroom activities of geometry, we articulate howchildren’s gestures embody the interactive nature of learning situationsand through which geometric knowing emerges and evolves.

BACKGROUND

Cognition and Embodiment

Over the past two decades, an increasing numbers of studies in cognitivescience and mathematics education have been published exhibiting theintegral role of the body in cognitive development (e.g. Nemirovsky &Ferrara, 2009; Núñez, Edwards &Matos, 1999; Radford, 2009). As a whole,this research severely undermines the metaphysical view of thinking, a viewof mind disconnected from the body and the world (e.g. Roth, 2010; Varela,Thompson&Rosch, 1991). By providing evidence for the irreducible role ofthe body in mathematical abstraction (thinking and learning processes), thisrecent work builds a foundation for the claims that knowing cannot be theaffair of a pure, independent, abstract, and disembodied mind. Such studiessupport the contention that cognition—at least in part—exists in bodilyrather than in mental form. Human cognition is embodied because bodily

GESTURE AND EMBODIED KNOWLEDGE 209

forms of experiences are extended to schemas and from their, by means ofmetaphorical and metonymical processes, to linguistic forms (Lakoff &Johnson, 1999). However, these linguistic forms of mathematics aremeaningful only because they activate the bodily forms from which theyderive and together with which they form the conception as a whole (Roth &Thom, 2009a).

Some research suggests that knowledge is best thought of as embodiedaction that continually emerges from interactive learning environments. Arecent edited volume examines the notion that mathematical representa-tions in use constitute the interface between embodied and culturalaspects of knowing and learning mathematics (Roth, 2009). By showingevidence of learners’ action and expression linked together in under-standing, mathematics education research shows how knowing is bodilyenacted in learning situations (Radford, 2009). For mathematical conceptsto develop, mental and physical actions and expressions have to beintegrated in children’s thought processes in complementary ways (Pirie& Kieren, 1994). Mental processes emerge not just as abstractions, asJean Piaget thought, but they require the body as a point of reference andas a source of experiences to think, learn, and develop.

The notion of bodily engagement as the necessary ground of children’slearning and thinking has long been discussed in educational discourse,especially after the emergence of theories according to which mentaldevelopment is the result of an abstraction by means of which bodilyoperations become mental operations (Piaget, 1970). But in Piaget’s work,bodily schemas are only stepping stones to formal thought rather thanintegral aspects. This approach is metaphysical, beyond the physical,because thought is for itself and no longer depends on the physical body.Thus, learners’ bodily engagement is, for many constructivist educators,only a transient path toward the final product of learning, that is,mathematical discourse (Radford, 2009). Bodily presentation has beentaken as a preliminary or subsidiary step for reaching abstract thought.Educational practices in classrooms continue to be grounded in thistraditional, metaphysical view of what it means to know. Consideringteaching and learning as the activity of transmitting–receiving knowledge,many teachers in mathematics classrooms heavily focus on children’swritten and spoken language as the outcomes of teaching and learning. Theknower’s bodily expressions and actions remain unrecognized when theonly means of accepted expressions of knowledge—e.g., on tests—are(written, verbal) texts. With this tendency, there has been a pervasive neglectof bodily dimensions of knowing and learning in the sciences and inmathematics (Rasmussen, Nemirovsky, Olszewski, Dost & Johnson, 2004).


This paper is based on a larger study that seeks to explicate embodiedfeatures of children’s mathematical knowing and learning in the elementarygrades. The study was conducted among second-grade students learninggeometry. In this paper, we pay special attention to children’s bodily actions,especially gestures of children’s geometrical thinking. Affiliating with theliterature on the embodiment of mathematical knowledge, we articulate whatknowledge children’s bodily engagements bring forth and how it isconnected to children’s imagination and abstraction in the development ofgeometrical thinking. Thus, children’s knowing that emerges from theirtouching and holding material objects, for example, is irreducible to verbaland written words—touching, taking hold, and gesturing tend to appear longbefore students learn to talk about scientific and mathematical phenomena(Roth, 2001). Knowing and recognizing the sound transcribed as “curve” isnot the same as knowing what it feels to touch a curved surface or to gesturea curve by means of an extended hand–arm movement. We discusschildren’s bodily actions as a necessary form of engagement for thedevelopment of geometrical concepts and illustrate learning situations withchildren’s gestures and the necessary engagements of children’s bodies inthe process of thinking geometrical ideas.

Bodies-in-Action Are Learning Bodies

There is an established and increasing focus in mathematics educationconcerned with gestures in mathematical situations and classrooms contexts(Goldin-Meadow 2004; Cook & Goldin-Meadow, 2006). Such researchexhibits (a) the situated and interactional nature of gestures in learners’articulation of concepts (Koschmann & LeBaron, 2002), (b) the impact andbenefits of mismatches between speech and gesture on children’s mathe-matics problem solving (Singer & Goldin-Meadow, 2005), (c) the nature ofspontaneous gestures as components of and inputs to conceptual integrationfor mathematical ideas, (d) the role of children’s gestures for bringing outimplicit knowledge, and (e) the gestural support in sense making in sciencelearning and teaching (for an extensive review of the literature, see Roth,2001). Understanding the relationships between speech and gestures, thesestudies show that gestures not only are communicative tools but alsoconstitute forms of reasoning and problem solving. For instance, Edwards(2009) suggests that conceptual blends of imaginary thinking and physicalaction necessarily take place in students’ mathematical problem solving andgestures instinctively emerge during those blends. Gestures, bodily experi-ences, andwords are but different, one-sided expressions of themathematicalconcepts that they metonymically denote (Roth & Thom, 2009a).


Other educational studies bring forth the notion that gestures are not onlycommunicative but tools for learning in classroom situations (for a review ofliterature, see Roth, 2001 or the special issue in Educational Studies ofMathematics, 70, 2009; e.g. Nemirovsky & Ferrara, Radford, or Roth &Thom). Still, there are few studies in mathematics education that closelyexamine the processes in which children’s gestures emerge and explicate theways in which children use gestures to express their thinking—through/intheir bodies. For example, in science education, there are studies that showhow symbolic gestures emerge from the hand–arm movements doinglaboratory work and how words, initially entirely absent, subsequently tookon an increasing load of communication and explanation (Roth, 2003). In theresearch literature, there are studies that conceive of gestures as expressionsof an inner mind (e.g. Alibali, Bassok, Solomon & Goldin-Meadow, 1999;Beattie & Coughlan, 1998; Hadar & Butterworth, 1997); in such studies,which include the body of psycholinguistic research on mathematics andgesture, we are not interested here. Such studies often explain the roles ofgestural engagement in children’s knowledge development; the evidence ofknowledge development is examined by means of textual mathematicsproblems. Understanding that verbal and written forms do not suffice toevaluate children’s knowing and learning, the present study depicts gesturesnot only as a means of knowing but also as a mode of knowing. Tounderstand the engagement of gestures in children’s knowing and learningof geometrical concepts, this study examines the following questions:

1. How do children’s gestures emerge and interact in learning situations?2. How do children’s gestures embody and develop their understandings

of geometrical concepts?

Framing our study with these questions, we inquire into how children’sgestures emerge, evolve, and become part of their geometrical under-standings in mathematics classrooms. We also attempt to look particularlyinto children’s gestures, sometimes without verbal language (speech) bygesturers associated in the thinking process so that we could understandmore in depth how children’s bodily engagements become cognitivelyactive to learn the concepts in classroom situations.

RESEARCH CONTEXT

To better understand the role of gestures in mathematical learning, thisstudy examines how children’s gestures emerge, evolve, and relate to


their thinking, interacting, and coping with complex ideas of geometry.We observed children’s gestures and interactions in second-grademathematics classrooms in a Canadian public school. The school islocated near the university and enrolls 430 students. These students comefrom ethnically, economically, and academically diverse backgrounds.The school is composed of 17 classrooms of K-5 students and offers adual track program of English (seven classrooms) and French Immersion(ten classrooms). This study was conducted in a second-grade classroomin the English program. There were 23 children (10 girls and 13 boys) inthe class. The children tend to speak English as their first language. Thechildren participated in 15 60-min lessons over 4 weeks. The lessonsincluded the curricular topics of three-dimensional objects, patterns, andmovements of the objects. Children were provided with three-dimensionalgeometrical artifacts to provide visual and tangible experiences with theobjects that they observed, touched, sorted, and manipulated (slid, rolled,and stacked) during classroom activities.

The regular classroom teacher is a dedicated teacher, who, at the time,was also the school’s vice principal. She is known in the school districtfor her leadership in curriculum and curriculum development as well asher work in evaluating curriculum materials for the BC Ministry ofEducation. This teacher was very interested in children’s development ofmathematical reasoning and welcomes our research team into herclassroom.

There were three researchers involved in this study. One researcher, thethird author of the paper, designed the tasks together with the teacher,taught on her own, team taught with the classroom teacher, and interactedwith the children during both whole-class and small-group sessions. Sheused gestures during her lessons to acknowledge this mode of knowingand to encourage children’s own gestures to emerge. However, she didnot, at any time, present any fixed gestures for particular concepts duringher lessons or give children direct instructions to use certain gestures forcertain occasions or concepts during lessons. In this way, the researcheras teacher did not demand but occasioned children’s gestures to emergenaturally and autonomously. The other two researchers (the first andsecond authors) observed and videotaped classroom activities.

To capture the dynamics of the children’s bodily engagement andenaction of geometry, we videotaped all lessons. Two video cameras wereused to record lessons and activities. When the teacher’s instructions weregiven to the whole class, only one camera recorded the classroom sceneas a whole while the other camera recorded a group of two children (Jadeand Sonia) to observe close up, any emergence, movement, or develop-


ment of gestures. When the children worked in groups, one of the twocameras followed a group for the duration of the activity. Video data weretransferred into QuickTime movie files (.mov) and shared among theresearchers and other colleagues in a research discussion group.

Studying the audiovisual data, we examined the embodiment,presentation, and development of children’s knowing in and throughtheir bodily engagements with objects. The researchers observed andinterpreted video files and discussed their interpretations, looking forintegrated themes of the children’s geometric conceptualizations in termsof embodied knowing and learning. The coding and analyzing of dataheavily depended on our collective observations and discussions of thevideo files since video resources provided integrated data of gestures innonverbal as well as verbal interactions in classroom situations. Weadhered to the method of interaction analysis (Jordan & Henderson, 1995;Roth, 2005), where groups of analysts interact to observe and theorize theinteractions among research participants. Collective analysis minimizesthe danger of subjectivism, as every observation and every explanationcome to be scrutinized within the research group.

In the first round of data analysis, the researchers identified andconjectured about possible themes concerning the relationships betweenchildren’s gestures and learning. The focus of data analysis in this initialround was on the emergence of gesticulation with and without speech.We analyzed gestures in terms of four categories: (a) gestures with notalking and no apparent communicative purpose, (b) gestures without talkbut with apparent communicative purpose, (c) gestures accompanyingtalk oriented toward others, and (d) gestures accompanying talk notdirected toward others. We coded gestures related to learning topics. Forexample, if when the teacher (or peer) was talking or questioning to thewhole class, a child made a gesture related to the teacher’s (or peer’s)speech without talking, we coded it as (a). If a child gestured withouttalking but interacted or communicated with the teacher or peer, we codedit as (b). If a child gestured and talked while directly communicating withothers, we coded it as (c). When a child gestured and talked by her/himself with no communication with others, we coded it as (d).

The unit of coding was an incident of gestural emergence in theindividual child, not each gesture. For example, if when the teacher wasexplaining some concept (e.g. a sphere) and a child made two gesturesduring the teacher’s explanation (making a round shape and rolling)without talking, we counted the incident as one case of the category (a),not two incidents of (a). If two children made gestures while the teachertalked, we counted them as two cases of the category (a). We did not


count any gestures if the teacher directly asked children to gesture, whichwas seldom since the teacher intended to occasion the emergence ofgestures naturally in this study. We tracked and documented the incidenceof gestures that were co-emerging with others’ (teacher or peers) gestures.The results of coding in this round are category (a) 65 (34.0%), category(b) 18 (9.4%), category (c) 104 (54.5%), and category (d) none (0%).During the coding, we also looked for gestures that emerged simulta-neously following the concepts that were explained or discussed by theteacher or peers. The numbers of co-emerging gestures were 24, 6, 10,and none in each category, respectively (Table 2).

The frequencies in Table 2 are subject to the following constraints:Some video did not capture all children in a particular scene. Therefore,the numbers do not reflect all the occasions of children’s gesturesthroughout the lessons. Another issue is that we questioned why theincidence of category (d)—gestures accompanying talk not directedtoward others—was not unlike our preliminary expectation. If childrenuse gestures to think, we expected that children would use gestures andtalk alone while engaged in activities of problem solving. We thought thatthese results might be due to the fact that most of the time the childrenwere engaged in interactive activities such as interacting with the teacher,group work, games, and discussions; therefore, opportunities for gesturingwith talking but no communication were seldom seen in this study.Sometimes, we could observe a few scenes that a child seemed to murmur

TABLE 2

Incidence of gestures in mathematics lessons

CategoriesNo. ofoccasions

No. of co-emerginggestures

Gestures withoutspeech

(a) Gestures with notalking and no apparentcommunication purpose

65 24 out of 65

(b) Gestures with no talkingbut with apparentcommunication purpose

18 6 out of 18

Gestures withspeech

(c) Gestures with talkingand communicating with others

104 10 out of 104

(d) Gestures with talkingbut no apparent communicationwith others

None None

Total 191 40 out of 191


and gesture, but it was captured from the distance and talking was notobvious. So we did not categorize it as (d) but (a).

After the first round data analysis, we selected samples of gestures infour categories and examined which gestures could be oriented towardchildren’s thinking or communicating with others. We interpreted that thegestures of (a) and (d) were more related to individual thinking processand (b) and (c) were more to interactive mode of learning. But followingVygotsky (1986), we understand those gestures directed toward othersduring communication as also expressions of developing one’s thought.Therefore, we did not separate the two categories as if they were entirelydifferent ways of learning or thought development in our datainterpretation. We then investigated scenes and episodes that representedthe emergence of gestures with and without speech more closely. In thisprocess, we developed the possible themes on the emergence of children’sgesture and learning and interactively observed the samples of gesturesagain. During the process, we questioned and changed the themes toreach integrated themes of the study.

In the third round of data analysis, we discussed themes within thescenes and episodes to look for the details of gestures in relation tothinking process. If gestures were more oriented toward children’sthinking for themselves than communicating and thinking with others,we sought to understand how gestures expressed thoughts and ideas. Ifgestures emerged in interaction and communication with others, weinvestigated when and how gestures were emerging throughout inter-active situations in order to understand the role of gesture in thinking andlearning process. The analyzed data and themes were also discussed forpeer debriefing with other researchers who do not have a stake in thisstudy.

In the following, we introduce short vignettes and detailed episodes asillustrative data to discuss the relationship of children’s gestures andembodied learning. These episodes demonstrate the scenes of children’sthinking through gestures and allow us to expand our discussion onembodied learning.

CASES OF CHILDREN’S GESTURES IN LEARNING GEOMETRY

This study was designed to understand the roles of bodily engagements,especially gestures in children’s learning and further to discuss the notionof embodied cognition in children’s conceptual development. In thissection, we support four claims: (a) there are gestures that support


children’s thinking and knowing, (b) some gestures co-emerge togetherwith peer’s gestures, (c) gestures cope with the abstractness of conceptualunderstanding, and (d) children’s bodies exhibit geometrical knowledge.By showing incidents of children’s gestures closely, these claims addressthe necessary question of how children’s gestures are integral in mentalprocesses of coordination and abstraction of concepts.

Gestures that Embody Children’s Thinking and Knowing

In this study, we observed many instances in which children’s gestures(43.4%) emerged in situations where children were not oriented towardothers. This observation prompted us to ask the following questions: Whydo children gesture when they do not orient themselves to others? Whatrole do such gestures play in children’s learning? We do know fromresearch that congenitally blind people gesture when they talk to others,even though they have never seen someone else gesture and even whenthey speak to other blind individuals (Iverson & Goldin-Meadow, 1998).This seems to suggest that gesturing is part of thinking and not just a wayof reading out thoughts to others. The following episode shows anexample from the second-grade classroom in which the children’sgestures emerged with neither speech nor communicative purpose todirectly interact with others. In this episode, we examine how thechildren’s gestures emerged and what roles they played in the learningsituation.

EPISODE #1. Prior to this lesson, the children worked in a variety ofother settings in the students generated distinctions about different shapedobjects as well as sorted and grouped three-dimensional objects accordingto several similarities and differences identified by the studentsthemselves (e.g. box shape, ball shape, pyramid shape, tube shape, etc.).In the previous lesson, children learned about movements (i.e. rolling,stacking, and sliding) with manipulating material objects. In this lesson,the teacher starts reviewing the movement of objects and converses withchildren why certain objects move in certain ways. While the students areexplaining why some objects slide and stack and/or roll, the teacher askswhat makes an object roll (turn 01). Then the teacher starts to conversewith the children about different faces of objects (turns 02 and 03). Alice,a second grader, responds to the teacher’s question with gestures of herown (Figure 1.1 and 1.2). She immediately and without speaking makes agesture that expresses rolling (Figure 1.1a and 1.2a) and again bends herright hand and makes a round shape with her hand along with the classroom


interaction (Figure 1.1b and 1.2b). The teacher asks the children what madethe sphere roll and Jade answers that it did not have any flat parts (turns 04and 05). Alice’s orients her upper body, face, and eyes toward the teacher atthis point (Figure 1.1a,b and 1.2a,b). Then the teacher asks again, “anythingelse about it?”, giving Molly a chance to respond. When Molly startsanswering the question, Alice’s bodily orientation turns toward her hand.Alice bends and puts both hands and fingers together on the desk andmakes a round shape (Figure 1.2c). Then she puts her two hands tight as shemoves them upward together (Figure 1.2d). Soon after she puts both handsdown and bends her right hand as if she was holding a round-shaped objectand lifts it again (Figure 1.2e).

01 Teacher: What makes it roll? Is it flat part or is it something else?

[((Alice rolls her two hands several times))]

We have flat surfaces like this, we

can put our hands just flat against it, and we have another faces like sphere with,

a02 Phil: a curve 03 Teacher: Yes, our hands go like this… [((Alice makes a gesture of a curve with

her right hand))] 04 Teacher: We also talked about a sphere.

What made a sphere roll? 05 Helen: It didn’t have any flat parts. 06 Teacher: Ok, it didn’t have any flat part

on it. Anything else about it? Molly? b

07 Molly: It does not have bumpy edges…(inaudible)

[((Alice moves her hand to make a shape with a round surface, moving her right hand up and down))]

c

Figure 1. 1 Alice’s gestures


Alice’s gestures of bending and rolling her hands articulate rolling androundness—this does not mean she has an explicit idea in her mind.Rather, rolling and roundness exist in the capabilities of her hands, which,in producing rolling and roundness, also express the memory of rollingand roundness (Maine de Biran, 1952). When the teacher asks if it is theflat part or some other part that makes things roll, Alice starts moving herhands. She continues to follow the teacher’s word “curve” by expressingthe formation of a round shape with her right hand. To think through whatmakes things roll, Alice uses both hands to gesture a three-dimensionalobject. Here, “thinking-through” exists in the movement of her hands, andwe have no evidence that any form of verbal thought accompanies thisform of thinking. It is certain that the object concept that has the shape ofa cylinder is in her hand: She literally has her hands on the concept. Hergestures emerge through interactions with the teacher and peers. Thinkingis handwork, according to the German philosopher Martin Heidegger(1982): “Man does not ‘have’ hands, but the hand embodies the nature ofman, because the word, as the realm of the hand, constitutes the nature ofman” (p. 119, our translation). Thus, in Alice’s gestures, we see thinkingas handwork literally occurring.

Given that there are may be discrepancies between children’s thoughts,words, and gestures (Goldin-Meadow, Wein & Chang, 1992), we cannotassure that what Alice’s gestures expressed was actually a cylinder. Norcan we make an assertion that her gestures presented that she understood

a b c

d e

She changes her bodily orientation from the teacher to herself and concentrates on her own thinking of concept and object. Figure 1. 2 The details of gestures and the changes of bodily orientation


the relationships between rolling and curved face. In fact, Alice’s gestureswere not straightforward to the teacher’s question at some points. Forinstance, she did not make a gesture of a curved face when the teacherasked what makes things roll and if it is a flat face or something else.Instead, she rolled her hands. Yet, even though her gestures were not theexact answers to the question, we can surmise that Alice was processingwith her body what is relevant to the concepts: the relationship betweenroundness and rolling. To develop conceptual understanding, a child putsindividual elements and encounters into groups of the un/related and dis/connected in their thought process, which is called the emergence ofcomplexes (Vygotsky, 1986). Building complexes is a preliminaryprocess for the development of concepts. Alice’s gestures show thatcomplexes of relationships or connections among ideas started emergingin her body. In/through her gestures, the complexes of what makes thingsroll is a curved face, not a flat face was emerging. She later develops around-shaped object in relation to concept of rolling (Figure 1.2). In thisprocess, gestures were essentially engaged in the development ofcomplexes and concept of rolling.

The change of Alice’s orientation is another important aspect of bodilyengagement in embodied thinking. When the teacher asks Molly toanswer the question, Alice responds to the situation by her own gesturingand body orientations. She starts gesturing, looking at the teacher, thenshe changes the orientation of her body from the teacher to herself andcontinues to gesture (Figure 1.2). Her gestures articulate embodied formsof thinking-through and conveying the response to the teacher’squestions, and later, her embodied thought process continues withoutmuch relation to the speech. She gestured another round shape (cylinder-like) instead of a sphere that the teacher mentioned. Alice explored ideas,in and through her body movements, concentrating on imaging a three-dimensional object with curved sur/face. In the process of imaging on herown, her body orientation changed and her hands were more activelyengaged in thinking. Her body was silent but active to process ideasanalogically, in and through her body, and to give shape to conceptsarticulated in the situation. At this moment of her thinking and learning,her body was more centrally present than any verbal or written inter/action.

From this episode, readers can learn how children’s bodily movementsare actively and necessarily engaged in building geometrical complexeseven before and without verbal expressions. The body is the expressionand thought, and it was more present than speech in Alice’s knowing andknowledge. Alice’s gestures express her bodily forms of knowing, here


nonverbal but active and for herself rather than in the shared public spaceof the whole-class interaction. To cope with the learning moment,children’s bodies are necessarily aspects of thought, different fromthoughts in the mind, but still expressions thereof. It is precisely thisform of thought that anything mental requires as its base (Maine de Biran,1952).

Gestures that Co-emerge with Peers’ Gestures

Our study shows that children’s gestures co-emerge through sharedorientations and interactions in the classroom. Some gestures (36% ofgestures without speech and 9.6% of gestures with speech) were co-emerging with gestures when concepts were articulated for others. Evenwhen children were not talking, there were many occasions of co-emergent gestures. For instance, in one small-group discussion, we foundchildren attempting to explain to peers the differences between atriangular prism and a rectangular prism. Before this lesson, the studentshad explored two-dimensional shapes as wells as the standard names forthem such as square, rectangle, triangle, and circle. Mark articulates anidea about triangular prisms for the benefit of his classmate, Brad. Theysit facing each other. While explaining, Mark gestures a triangular shapeusing both hands. He puts the two thumbs together signifying thetriangle’s base and the rest fingers attached signifying the other two sidesof the triangle. Then he raises his hands up, retaining the handconfiguration. Without talking, Brad simultaneously makes gestures thatcorrespond to Mark’s talk and gestures. With his right hand, he draws atriangle in a lower position and some straight lines up from it in the air.His gestures are seen to convey the same content as Mark’s speech.However, Brad’s gestures may not have the purpose of conversationalcommunication, since the gestures emerged almost simultaneous withMark’s. What does the co-emergence of Brad’s gestures suggest in termsof embodied learning? We suggest this answer: The gestures express“once-occurrent uniqueness or singularity cannot be thought of, it canonly be participatively experienced or lived through” (Bakhtin, 1993, p.13). Being situated in classroom communications, children participativelyand live through the shared production of concepts, which, as any higher-order cognitive form, initially exist in and through social interaction(Vygotsky, 1978). The shared boundaries of learning environmentsdevelop the connectivity of knowing at collective levels. Children’sgestures as embodied cognitive action are situated and contingent, relatedto the way children relate to others, to things, and to themselves. To


discuss these notions more in detail, we illustrate a case of co-emerginggestures through learners’ interactions in the following Episode #2.

EPISODE #2. In previous lessons, the children developed relationshipsbetween the attributes of objects (cube, prism, pyramid, cylinder, andsphere) and their movement(s) (slide, stack, and/or roll). The challenge ofthis lesson was for the children to predict and justify what the objects’movement(s) might be when they set on an incline plane. The childrenworked in small groups and explain as well as justify which objects theythought might slide or roll down the incline plane without handling theobjects. In the earlier part of the lesson, the teacher talked about corners(vertices), faces, and edges. At this point, the children still use these termsincorrectly, for instance, as seen in this episode, they used the word“point” to denote an edge.

The teacher comes to Jade and Sonia and asks what will happen whenthey put a rectangular prism down the plane. This question challenges thechildren to think through the relationship between the movement andshape through one particular object. Jade, who is standing beside Sonia,responds to the question. Without showing the real object, Jade needs toexplain what would happen to the rectangular prism when it is sent downthe incline plane. She says it will slide because the faces of the object areflat. As Jade explains this, she also uses her hands to gesture, articulatingthe shape of a flat base and a point on the top of a rectangular prism. Sheopens both of her palms horizontally and then joins them together andmakes a shape of an angled hill (Figure 2a, b). She continues by sayingthat curviness cannot make objects slide. At this point, she bends bothhands, putting the right hand in an upper position and her left hand on thedesk (Figure 2c). By joining her flat palms together in an angled position,she explains how the characteristics of rectangular prism—i.e., theflatness of the bottom face—would make the object slide. In this episode,she thinks through the difference between flatness and curviness andexpected movements.

Through verbal and gestural responses, Jade expresses the shape of arectangular prism. Here, geometrical thought is not abstracted but existsin and through tactile and kinesthetic actions and imagination. Theseallow children to develop their own images of shape and motion. It isprecisely because geometrical thought is not abstracted that children canlater “apply” in the real world any geometry word: It is only whenchildren are taught words without accompanying bodily experiences thatgeometrical words make no sense to them. Based on these ideas, wefurther question how a child without speech coordinates geometrical ideas


through gestures in interactive classroom situations. In this way, wedevelop a better understanding of the role of gestures in coordinatingideas. In this episode, we investigate more closely Sonia’s responses.While sitting next to Jade, Sonia also begins to gesture. While Jade talksand makes the gestures of flatness, pointy part, and curviness of shapes,Sonia also produces the corresponding gestures.

Sonia, without actually speaking, succeeds in coordinating the imageof shapes by means of gestures. While Jade is speaking and making agesture of a surface without curvature, flattening her both palmshorizontally, Sonia attaches her palms together vertically (Figure 2a).When Jade says “it gets a point” while making a point by putting herhands together in an acute angle, Sonia stretches her right hand and sticksout the right thumb and then wraps it with her left palm as if she wantedto distinguish the sticking-out thumb (Figure 2b). For the curved face,Jade bends her right hand and Sonia bends and folds her hands together

01 Helen: Because it’s flat [((Helen makes a flat face with her two hands))]

[((Sarah puts her hands together palm to

palm))]

a 02 Helen (continues): and it gets a point.

[((Helen makes a triangle shape to express an edge of prism with her two hands))]

[(( Sarah sticks her right thumb out and puts her

left palm around the right thumb))]

b 03 Helen (continues): If it’s curved, then it’s just

going like this. [((Helen draws a curve moving her hand and later rolls her hand down))]

[((Sarah makes a curve shape with her two

hands))]

c

Figure 2. Sonia’s co-emerging gestures in group interaction


(Figure 2c). In her gesture, Sonia expresses flatness, curviness, and theangled part of object that can be seen as an edge. During the entireprocess, Sonia does not speak. She only orients her body toward theteacher and moves her hands to respond to the situation. Through herhands, the shapes of objects are made present—made present again,represented—which exhibits a bodily form of thinking. While nobody cansee into her mind, such bodily forms in fact are all that human beingshave available to communicate (about) geometrical concepts. It is only ina highly idealized world that space—and therefore geometry—is prior toand therefore independent of (the objects of) human experience in theway both Kant (1956) and Piaget (1970) thought.

During this process, it is important to notice that even if Jade and Soniaresponded to the same question, their ways of responding differed. Thismeans that Sonia is actually coordinating other possible thought formsrather than acting without thinking. Sonia’s gestures are creative andimmediate actions to connect her to the moment. She is not a passivelistener, but active and engaged in abstraction of the relationship betweenthe shape and movement of a certain object, which Bakhtin (1993)explains as participative thinking. In the nature of participation, thinkingis nonlinear and emergent, and thus, gesturing cannot be a simple copy ofsomebody else’s or representation of learned symbols. The bodyparticipates in abstracting the ideas unfolded in the interaction imagi-natively and spontaneously. Therefore, Sonia’s gestures could co-emergewith Jade’s, not as mimicking but as creative presentations of her ownknowing. From this instance, we may conclude that geometrical thoughtinvolves children’s gestures as real-time actions of imagination tocoordinate geometrical meanings.

The nature of mathematical activity is collective as well as individual.Vygotsky (1997) argues that any function of a child’s thought develop-ment first appears in interaction with others. That is, before we can thinkgeometrical ideas in private, we have already encountered in public thepossibilities for certain images and verbal thought forms. Through thesocial and individual abstraction and reflection, mathematical meaningsemerge collectively among learners. And with the intersubjective,interactive nature of learning, children’s gestures co-emerge as childrenarticulate themselves for the purpose of articulating for others. Theepisode reveals that the shapes emerge in and through social interaction.The emergent nature of imaginative, physical actions is possible whenlearners are situated and embodied in interactive and collective situationswhere the outcomes of interactions are neither linear nor predictable. Jadeand Sonia’s bodies are orienting in the same way, physically (in space)


and metaphorically (toward the mathematical object). The children’sbodily movements synchronize to follow the ideas that unfolding in frontof them in the once-occurrent here and now of the present lesson. Beingsituated in the shared situation, their bodily orientation (face, eyes, andupper bodies), speech, and gestures are indistinguishable from theirthinking and knowing. Immediately responding to the situation by pairingwith Jade’s speech and gestures, Sonia’s body is oriented to and coupledwith others. None of these bodily inter/actions can be understoodindependently of thinking and learning in this situation, and thinkingand learning cannot be understood independently of the interactions.Children as interactive learners create intersubjective, collective con-sciousness, which brings forth shared bodily orientation and participationand the emergence and connectivity of knowledge among learners (Horn& Wilburn, 2005; Roth, 2007). In other words, intersubjective knowingemerges whenever individual learners reciprocally assimilate the actionsof the other in socially interactive situations (Steffe, 2003). Jade andSonia’s gestures in this episode are evidence of the emergence of bodilyand imaginative communication, which is inherently collective andcreative.

Gestures that Cope with the Abstract

Conceptions evolve from and exist in and through primary experiences asthey become connected and mutually mediate and stabilize one another(Roth & Thom, 2009a). This is so because the relation between cognitionand the world presupposes an interlacing that can only be achieved in aliving body, in the flesh (Merleau-Ponty, 1964). Any practically relevantconception exists in and through embodied forms. If it were not like this,we would not know that a particular sensation is related to a geometricalconcept, on the one hand, and we would not know how to act in the worldif our geometrical understandings were to be pure thought. Based on herintuitive understanding of this idea, the teacher provides the children withopportunities to experience three-dimensional objects by sensual means.Through the experiences, the teacher provides the children withopportunities to develop their knowledge of geometrical objects invarious modes such as the language of objects (e.g. rectangular pyramid,sphere, etc.), diagrams or figures (drawings of objects), and classificationsof material objects (grouping). In this process, the children are expectedto relate different ways of knowing to one object, for example, a cube ontheir hands, the diagram of cube drawn on paper, and the sound that istranscribed in the conventions of the International Phonetics Association


as “kju:b”, but which English speakers here as an instance of the word“cube”. To help students connect their experience of three-dimensionalobjects to other forms of representations, the teacher engages them invarious activities of naming and name-making, grouping and matchingcards of names or diagrams, and explaining physical demonstrations withor without material objects. During activities, the teacher also challengeschildren to provide cards of objects that they have not seen in three-dimensional form. This allowed the research team to observe howchildren’s gestures were engaged to cope with a new and abstractsituation. Throughout children’s problem solving with those challenges,an ongoing cycle of embodied learning took place in the dynamic balanceof different forms of knowing. In other words, to understand and solve anew, abstract task, children had to integrate their knowledge andexperiences of objects and also applied them into an unknown situationwith creative imagination.

In this study, we come to understand that for the abstraction ofgeometrical objects, the children’s bodily activities became grounds forconstituents of reflecting on geometrical concepts. For children, gesturingwas a way of coping with the challenge of unknown objects and ofbringing forth a dynamic balance of abstracting and acting on learningconcepts. When children encountered new ideas, their gestures wereemerging and organizing their thinking in visible and concrete dimen-sions. In observing this process, we have no evidence for the separation ofthinking and communicating, which are two processes that in fact appearto mutually stimulate each other (Vygotsky, 1986). As the conceptsbecome more abstract and complicated over time, children’s handmovements also embody the complexity of thinking and expressing overthe concepts. The following Episode #3 shows an example of children’sgestures in the process of thinking in a new situation.

EPISODE #3. To understand the relationships between the types of facesand the movements of objects, children participated in various tasks. Theylearned the different parts of objects (faces, vertices, edges, etc.) and theprocesses of rolling, stacking, and sliding. They observed these processeswith real objects. They predicted which objects would roll, stack, andslide, and they explained their rationale behind their predictions. Theyalso made their own names and learned standard geometric names forfigures of material objects. In this episode, children are expected toexplain the relationships between geometrical objects and their associatedmovement(s) (rolling, stacking, and sliding) by applying their conceptsthey already learned.


In this situation, the children are playing a card game that requiresthem to name objects on cards and explain their thinking of how an objectwould move without manipulating real objects. About 15 cards weregiven to each group of children. Each card contains a drawing of differentgeometrical objects. Most of objects were observed and tested in theprevious classes with artifacts. However, there are also several newobjects such as a hexagonal prism, an ovoid, and a half sphere thatchildren have not investigated previously. When it is Jade’s turn, shepicks up a card and flips it. She looks at the card featuring a half sphereand she says, “Can it roll? Actually, it can roll and slide.” Jade startsexplaining by using her hands. She shows how the object would slide(turn 01, Table 3) as well as roll (turn 02). However, Sonia does not agreewith Jade (turns 03–07). Sonia starts explaining her ideas with gestures.

When Sonia and Jade are challenged by the object that they have notseen before, their knowledge about relationships of shape and movementis reflected to the newness of the object. Their gestures seem to become

TABLE 3

Sonia and Jade talking about half sphere

Turn Verbal and gestural interaction

01 Jade: If you put this way, it’s gonna all slide, [((Jadeslides her hand))]

02 but if you put on that part, it’s gonna start rolling likethis [((Jade rolls her hand))]

03 Sonia: Jade, you know what?04 With this, it couldn’t roll because roll,05 it goes, it repeats. [((Sonia rolls her hands on the desk))]06 And this one, it can’t because it’s roll for a part07 [((Sonia twists her hands once))]

and then the face that is flat goes and stops. [((Sonia rollsher hands once and stops the movements))]

08 Jade: But the bottom side, it would slide09 but this side, it would roll because it is part of sphere.10 Sonia: Yes, but it’s a half so it’s flat in the middle, and it’s

curve. [((Sonia moves her hands as if she cut an object in half))]11 If you roll it, it will stop because it will go,12 so it’s like this, [((Sonia rolls her hands))]13 If you roll it like this, it will go one time, [((Sonia puts her two

hands together and makes a bowl-like shape and slantsher upper body as if her body moved along with the object))]

14 then it will stop.[((Sonia’s body gets more slant and stops moving))]


central in the effort of coping with the abstract image. To think throughthese new ideas, Sonia uses gestures—much like James Watson andFrancis Crick were moving about cardboard shapes to think through ideasabout the structure of DNA. Sonia first explains that rolling meansrepeating (turns 04–05), crossing her right hand over the left hand(Figure 3a). Then she starts to explain why she thinks that a half spheredoes not roll by talking about a flat face. She puts her two palms down tothe desk first and then puts her left hand on the desk vertically and movesher right hand from up to down as if she was slicing something on verticalline (Figure 3b, c). To convince Jade, who does not agree with her easily,Sonia elaborates her explanation by adding more gestures (Figure 3d–g).She explicates the relationship of a flat/curved face and the movements itenables. Together with her talking, she makes gestures of twisting androlling with her two hands and arms (Figure 3d). Then she curves andjoins her hands to make a bowl-shape figure (Figure 3e), and she startsmoving her joined hands, arms, and her upper body as if she were holdingthe object and moving along with it (Figure 3e, f). When the object stops,her body also stops moving (Figure 3g). With the gestures, Soniaarticulates why a half sphere cannot roll. The nonpresent object takes on avirtual presence in her bodily movements.

The development of Sonia’s gestures in this episode exhibits thecomplexity of Sonia’s thought in a new situation. In and through hergestures, she is organizing and presenting her thoughts in visual form.

a (turn 04, 05) b (turn 6,7) c (turn 10) d (turn 11)

e f g(turn 12) (turn 13) (turn 14)

Figure 3. Sonia’s explanation of a half sphere


Because translation means treason, it would be inappropriate to saythat these thoughts also existed in verbal form unless there is explicitevidence for such forms to be present and relevant. When she facesthe novel object, that is, the half sphere, she mobilizes bodilyknowledge of rolling, of which curvy face rolls and flat face goesand stops. To argue for her ideas, she evolves accurate and logicalforms of gestures of half sphere and its movement and helps Jadeunderstand that a half sphere cannot roll. Sonia features the shape andmovement of the object with her hands as if she were holding andmoving material objects. Sonia coordinates, anew, between two faces(curve and flat) and two motions (rolling and not rolling) that arealready known to her (e.g. curve and rolling), which leads to ananticipation about how a body must move (will not roll beyond thehalf ball part of it). Without operating on the real object, she stillsolves the problem in a visual manner. Her gestures emerge to reflectideas in this new situation. Her thinking develops in and through hergestures, and her gestures further develop her thinking. Her gesturesconstitute the thinking with and about shapes and motion.

Geometrical Knowledge in Children’s Bodies

As the geometrical concepts became more complex along the course, thechildren’s gestures also became integral and more complex. Thus, at thebeginning of the study, children’s gestures were simple and fragmented.They used gestures disjointedly to explain ideas such as round, flat, orroll. For example, they bent or rolled their hands once in the context ofwhich sphere could roll, neither stack nor slide because it did not haveany flat faces. Later, children’s gestures developed to cope with complexrelationships between concepts. The development of Sonia’s gestures inthe Episode #3 is an example of this idea. Before Sonia encountered thechallenge of half sphere, the task was to answer what rectangular prismswould do. Sonia earlier explained that a rectangular prism would slidebecause it had all flat faces. She also explained what rolling meant. Tosolve the problem of half sphere, Sonia was connecting the concepts ofcurve and flat and rolling and sliding (see Table 4).

These data indicate that Sonia coordinates, in a new way, two faces(curved, flat) and two movements (rolling and sliding) that were alreadyknown to her. This led to a novel anticipation about how the half spherewould move (will not roll beyond the half ball part of it). In and throughher body, she articulates for herself as much as for others that rollingneeds to repeat without stopping and flat face does not make things roll.


Therefore, she explains that a flat face on half sphere does not allow theobject to roll but that it will stop at some point. Each individual gesture(e.g. round, flat, rolling, sliding) is combined in the ideas of three-dimensional objects (e.g. sphere, rectangular prism) and further in therelationships among object, shape, and movement (e.g. sphere-roundface-rolling and rectangular prism-flat face-sliding). The child’s gesturestherefore constitute and exhibit a form of conceptual development (seeFigure 4).

There have been suggestions that children have the ability of formingmental images in their abstraction of concepts and they actually processmental operations of those images to solve mathematical problems given(Steffe, 2002). But anything that may occur in the privacy of our mindsinevitably is the result of social relations (Vygotsky, 1978). What thechildren in our study might do in private and inaccessible to observationinevitably has arisen from the ways in which they have related to others.These ways always are public and available to the senses—no educator orpsychologist would use extra sensory perception and communication as avalid explanation of learning. As Sonia articulates the movements of a halfsphere that she had never seen or touched before, imagining is enactedwitnessably in and through her body movements. Her body developed andembodied the geometrical concepts (shape and movement).

TABLE 4

Conceptual development through speech and gesture

Turn Sonia’s speech Sonia’s gesture Possible concept

05 it goes, it repeats. She rolls her hands. Roll meansto repeat.06 And this one, it can’t

because it’s roll for a partShe twisted herhands once.

07 and then the face that isflat goes and stops.

She rolls her handsand stops the movements.

Flat face doesnot roll.

10 Yes, but it’s a half so it’sflat in the middle, andit’s curve.

She moves her hands asif she cut an object in half.

A half spherehas a flat facein the middle;

11 If you roll it, it will stopbecause it will go,

12 so it’s like this, She puts two hands togetherand makes a bowl shape.

therefore, itcannot roll.

13 If you roll it like this,it will go one time,

She moves her body as if shewas holding a half sphere.

14 then it will stop. Her body stops moving.


Our video materials show how children actively think and learn therelationships of objects, shapes, and movements in and with their bodies.Without doubt, the neurons of the mind are somehow involved. But whatmatters to mathematics educators is what children such as Sonia makeavailable to others, and these are body movements, hand and armgestures, and words. There is nothing abstract about concepts if werecognize how they are constituted across these varying bodily expres-sions. There is no need to draw on theories of the invisible, no need forinaccessible mental images and schemas, as everything required forcommunicating concepts and solving problems is enacted and madeavailable by bodily means. There is no doubt that Piaget over-intellectualized knowing (Merleau-Ponty, 1945/2002; Varela et al.,1991) and thereby merely reified another metaphysical theory of knowingthat is of a different matter than the material of this world. Once weaccept the embodiment hypothesis, we are actually in a position toappreciate that mathematics emerges “as meaningful structures for usmainly through the bodily experience of movement in space, manipu-lation of objects, and perceptual interaction” (Núñez et al., 1999, p. 51).The patterns of mathematical encounters order children’s perceptions,

3-dimensional objects characteristics of object movement

Spheres, cylinder Round faces Rolling

prisms (triangular, rectangular, hexagonal)

Flat faces, edges Sliding, Stacking

Pyramids (triangular, rectangular, hexagonal)

Pointy tops (apex) Sliding

Roundness-Rolling

Flatness-Sliding

Flatness-sliding

Sphere

Rectangular prism

Pyramid

Task: how will half spheres move? Roll? Stack? Slide?

half sphere

Flatness & roundness sliding? Rolling? Bodily coping and

problem solving

Figure 4. The relationships among characteristics, shapes, and movement


conceptions, and actions and the whole process of cognitive experiencesexpand children’s dynamic articulation of mathematical concepts.

RETHINKING KNOWING AND KNOWLEDGE IN EMBODIED LEARNING

This study was designed to investigate how children’s bodily experiencesunderpin their knowing and learning of geometrical concepts said to be tooabstract for their age. First, we found that children’s bodies (bodilyorientations and gestures) constitute an integral part of knowing, thinking,and learning supporting the appropriate of geometrical concepts before the agethought possible. Children occasionally used gestures without speech whenthey explored, processed, and expressed their own ideas. We exemplify thesefindings in the study of three children. For example, Alice’s hands, withouttalking, internalize the meaning of concepts (e.g. a round face) through and inher body. Sonia thinks about how objects move in and through her body.When the children encountered new situations, their bodies become resourcesfor actively coping with the problems. As seen in Sonia’s movements,children’s bodily coping brings forth the dynamic, dialectical balance withingeometrical knowledge. Being engaged in abstracting the knowledge ofobjects, gestures become constitutive forms of geometrical knowing.

Second, the children’s co-emerging gestures allowed new concepts tobe enacted for themselves and others and, thereby, for new concepts tobecome reflexive objects available to individual and collective inspection.Brad’s and Sonia’s co-emerging gestures in Episode #2 suggest thatchildren’s living, interactive cognition makes possible the emergence ofcollective knowledge in learning situations. As individual sense makingand collective knowledge emergence are co-implicated in interactivelearning situations (Davis & Simmt, 2003), children’s bodies as learningsystems are necessarily embedded in individual and collective ways oflearning geometrical concepts. During the group interaction, the wholescene of co-gesturing became the collective understanding of geometricalobjects among the learners. Because children communicate for others,their actions are also shared: “consciousness-for-myself”, as the etymo-logical root suggests (Lat. con-, with + sciēre, knowing), always is a“practical consciousness-for-others” (Vygotsky, 1986, p. 256). Thereflexive, enactive bodies resonate with intersubjective understanding ofgeometrical concepts among the children. Accordingly, their knowledgeis shared, co-emerges, and co-develops. In other words, their knowledgegrows through their connected, interactive bodies and minds in theenvironments where they are situated. This study highlights that


children’s bodily movements are cognitive actions to orient and processthinking and knowing of geometrical concepts. The body as the lynchpinof thinking and knowing concretizes, reflects, and embodies the livedexperiences and interactions of shapes and movements of objects. As thechildren’s bodies are bounded together with the objects and phenomena infront of them, their imagining and reflecting on them are also situated,intersubjective, but in creative ways.

Requestioning of the scheme of body is indeed relearning of ourknowledge of the world “as we are in the world through our body and weperceive the world with our body” (Merleau-Ponty, 1945/2002, p. 239). Therecognition of the body in one’s thinking and knowing of conceptschallenges the received views of knowledge in learning and teaching.Knowledge development is not a result of some invisible construction butthe result of social relation. Knowledge only exists as it is expressed in andthrough action. This further emphasizes that knowledge is not in verbal andwritten languages alone but also bodily actions as seen in which children’sbodies embodied and presented the shape and movements of objectssometimes without speech. Sonia’s knowing of half sphere cannot bereduced to only her speech. Her knowing and knowledge exists also throughand in her gestures. Embodied cognition suggests whatever we recognize asrational, scientific, or mathematical is fully embedded in our bodily actions(Nemirovsky & Ferrara, 2009). This notion of embodied knowledgechallenges received ways of theorizing mathematical and scientific knowingwhich have missed important aspects of practical, bodily thinking in theprocess of concept development. When knowing takes place fromcoordination of mental and bodily interactions among learners, knowledgeneeds to be understood and developed in and through those interactions. Thisapproach allows us to rethink knowledge in terms of consciousness, whichinherently is consciousness with and for others.

Despite two decades of research on the embodied nature of cognition,constructivist perspectives continue to emphasize abstraction of knowledgefrom the physical engagement with the world (e.g. Battista, 2007; Mulligan,Mitchelmore & Prescott, 2005; Outhred & Mitchelmore, 2004) and inessence a theoretical articulation of metaphysics. In this way, constructivismorients our attention away from the body, concerned as it is with theconstruction of mental entities and representations. In this study, we exhibitthe central role of the body in children’s knowing and learning to encourageteachers and curriculum designers to refocus teaching methods andevaluation/testing. Once we acknowledge the body as the seat of knowledgeitself rather than as a stepping stone to abstractions, it is possible to organizeteaching differently. In doing so, brings forth the possibility of inquiring into


such questions as, “What might we do to recognize and understandchildren’s knowledge expressed in modes other than speech or writing?”Such a question prompts us to look into alternative ways of understandingand promoting children’s learning in classrooms in addition to their verbal orwritten performance as knowledge development. Besides verbal articulation,reports, or answers to written tests, some knowledge of science andmathematics such as modeling, performing, problem solving, or doingpractical work are bodily oriented knowing, whose process cannot bepresented only through verbal and written contexts. For instance, learnerssolve problems in science and math project through bodily interactions suchas designing, drawing, measuring, or manipulating objects. Learners maycarry out practical work to understand the unfolding scientific phenomena infront of them. Much of their knowledge of science and mathematics ispromoted and exhibited through their bodily engagement and movements,not only verbal speech or written products. However, it is not commonpractice for mathematics teachers to pay attention to children’s bodilyengagement as a way of knowing. Rather, assessments are done using formalrepresentational means, and although what students provide on a written testor verbally as a response to a teachers’ questions are certainly ways ofunderstanding students’ knowledge, these forms of assessment are limited inthe opportunities they afford toward understanding the complexity ofchildren’s cognitive development and restructuring forms of teaching andlearning by engaging bodily experiences. As children think, develop, andexpress knowledge through their bodies, their bodily engagement needs tobe realized as integral to student learning. Their bodies are necessarilyengaged in coping with the abstractness of knowledge. Their bodies embodythe knowledge of science and mathematics and become part of knowingitself. Thus, despite the ever-increasing research on embodied knowing,there continues to be a need for developing strategic ways of understandingchildren’s bodily engagement and expression as it exists in classroomsettings as embodied cognition.

Throughout this study, we articulate the central roles of children’sbodily engagement, especially gestures in classroom settings. Throughexhibiting the cases of children’s gestures, we substantiate how gesturesplay an integral role in the children’s geometry learning. However, wealso encountered some issues in this study. First, we could not exhibitdistinctive cases to explain how an individual learner used gestures whens/he was involved in problem solving process, which might provide usmore descriptive details of gestural engagement in thinking anddeveloping ideas. Throughout the study, the children engaged in aninteractive mode of learning with talking-out-loud strategies, that is, the


children were required to explain their thinking out loud most of the time,or they played interactive games. In other words, the children engaged inverbal interactions most of times. We designed the lessons in such waysbecause one of the study intentions was to understand children’sembodied cognition in collective learning environments, and therefore,the children seldom worked on tasks alone. This instructional strategymight have limited us to examine in depth how gestures developindividual thinking and cognitive process as children are engaged inproblem-solving tasks without communicating with others. To probe theprocess of cognition developed by gestures, there needs to be furtherresearch to bring forth individual problem solving tasks and interpretchildren’s gestures throughout the tasks over time. And yet, this approachalso needs to be approached carefully with realization of collective natureof embodied cognition. Second, we did not look into children’s gesturesto evaluate their knowledge of geometry in this study. Therefore, therestill remains a challenge that we need to develop pedagogical strategiesfor using children’s gestures to evaluate their knowledge development.When we overcome this challenge, there may be more practice ofengaging and using children’s bodily engagements in classrooms. Futureresearch with teachers might allow us to identify means of evaluation thatare both ecologically valid and feasible within the temporal constraintsthat schooling imposes. But we do know that failing to pay attention towhat learners’ bodies know and present as collective forms of knowing,we may not understand the whole process of children’s thought andknowledge development.

ACKNOWLEDGEMENT

This research project was supported by several grants from the SocialSciences and Humanities Research Council of Canada. We are grateful toLilian Pozzer Ardenghi who helped us in the videotaping. We want toexpress our gratitude to the second-grade students, their regular teacher,the teacher intern, and teachers on call who assisted us during the weeksof our study.

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Mijung Kim

Natural Sciences and Science Education, National Institute of EducationNanyang Technological University1 Nanyang Walk, Singapore, 637616, SingaporeE-mail: [email protected]

Wolff-Michael Roth and Jennifer Thom

University of VictoriaVictoria, BC, Canada


children’s gestures and the embodied knowledge of geometry

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