Classroom mathematical practices and gesturing

Journal of Mathematical Behavior 23 (2004) 301–323

Classroom mathematical practices and gesturing

Chris Rasmussena,∗, Michelle Stephanb, Karen Allenca Department of Mathematics and Statistics, San Diego State University,

5500 Campanile Drive, San Diego, CA 92182-7720, USAb University of Central Florida, USA

c Purdue University, USA

Available online 29 July 2004

Abstract

The purpose of this paper is to illustrate a methodological approach for empirically investigating the functionof gesturing in the collective development of knowledge. We extend the earlier work of Stephan and Rasmussen[Stephan, M., & Rasmussen, C. (2002). Classroom mathematical practices in differential equations.Journal ofMathematical Behavior 21, 459–490] who analyzed classroom discourse and symbolizing to document the emer-gence of six classroom mathematical practices over the course of 22 days of instruction on first-order differentialequations. We complement and extend this previous analysis by re-examining the same data for gesturing andcoordinate this analysis with the evolution of the classroom mathematical practices as they developed in this par-ticular community of learners. Our illustration of the methodology we developed suggests that (1) gestures andargumentation can function as a unit that supports the establishment of one or more taken-as-shared ideas, and (2)that a gesture/argumentation pair that develops while establishing one practice can change function to support theestablishment of ideas embedded in other classroom mathematical practices.© 2004 Elsevier Inc. All rights reserved.

Keywords: Gestures; Practices; Learning in social context; Argumentation; Differential equations

1. Classroom mathematical practices and gesturing

Over the past decade an increasing number of researchers have developed theoretical and pragmaticaccounts of learning and teaching from sociological points of view. Examples of such studies in whichlearning is viewed as a social process include the work ofBall and Bass (2000), Cobb and Bauersfeld

∗ Corresponding author. Tel.: +1 619 594 7241.E-mail address:[email protected] (C. Rasmussen).

0732-3123/$ – see front matter © 2004 Elsevier Inc. All rights reserved.doi:10.1016/j.jmathb.2004.06.003

302 C. Rasmussen et al. / Journal of Mathematical Behavior 23 (2004) 301–323

(1995), Lave and Wenger (1991), Rogoff (1990), Saxe (1991), Stephan, Bowers, Cobb, and Gravemeijer(2004)andWenger (1998). A common thread among this work is a focus on how communities of learnersparticipate in and develop various practices. What constitutes a practice and how researchers can documentthe evolution of such practices is an active area of research. For example,Wenger (1998)describes howthe notion of practice, which eschews traditional dichotomies such as those between thinking and doing orbetween abstract and concrete, includes all the explicit and the tacit means by which say, claims processors,are able to do their job. Focusing specifically on the mathematics classroom,Cobb and Yackel (1996)describe a classroom mathematical practice as the taken-as-shared ways in which a classroom communityreasons, symbolizes, and argues. Common to these descriptions is the view that practices involve theemergent and negotiated social production of meaning. Methodological approaches for documenting thecollective growth of mathematics understandings and views about mathematics teaching and learning arealso beginning to be developed (e.g.,Bowers & Nickerson, 2001; Cobb, Stephan, McClain, & Gravemeijer,2001; Cobb & Whitenack, 1996; Stephan & Rasmussen, 2002; Yackel, 1997).

Consistent with the emergent perspective as described byCobb and Yackel (1996), our research ap-proach views mathematics learning as both a collective and individual process. In analyzing the collec-tive, or social aspects of mathematics learning, the vast majority of work within the emergent perspectiveparadigm has focused on how classroom discourse, tool-use, and/or forms of symbolizing contribute tothe constitution of classroom mathematical practices (e.g.,Gravemeijer, Cobb, Bowers, & Whitenack,2000; McClain & Cobb, 2001; Rasmussen, 1999; Stephan et al., 2004; Stephan & Rasmussen, 2002).An important connection to this line of research is that knowing, rather than being located entirelyin the mind, is distributed across activities involving tools, utterances, and symbols (Pea, 1993). Anotably underdeveloped aspect of this line of inquiry is an examination of how gesturing might con-tribute to the collective development of meaning. In keeping with theories of distributed intelligences,we cast learning as distributed not only across tool-use, discursive activity, and symbolizing, but alsoacross gesturing activity. Along withMeira (1998), who describes learning as reflexively related totool-use, learning for us is also inseparable from gesturing activity; the two develop simultaneously(cf. Lakoff & Nunez, 2000; McNeill, 1992). Thus, focusing on the function of gesturing in the col-lective development of meaning can be an important, yet often neglected component of mathematicslearning.

The purpose of this case study is to illustrate a methodological approach for empirically investigatingthe function of gesturing in the evolution of the taken-as-shared ways of reasoning in one differentialequations class. In other words, our focus is on how to account for the activity of gesturing as it relatesto the social construction of meaning. We extend the earlier work ofStephan and Rasmussen (2002),who analyzed classroom discourse and symbolizing to document the emergence of six classroommathematical practices over the course of 22 days of instruction on first-order differential equations. Thisprevious analysis did not include gesturing. We therefore extend this earlier analysis by examining thesame data for gesturing and coordinate this analysis with the evolution of the classroom mathematicalpractices as they developed in this particular community of learners.

The significance of this research is twofold. First, we offer a methodological approach for analyzinggesturing as it relates to the collective construction of meaning, complementing the majority of gesture-related research that has tended to focus on classifying types of gestures and/or on function of gestures forindividuals. Second, the analysis contributes to ongoing design research efforts in differential equations,making explicit gesturing activity that has the potential to inform the work of teaching and promotestudents’ conceptual growth. As such, this research enhances the differential equations instructional

C. Rasmussen et al. / Journal of Mathematical Behavior 23 (2004) 301–323 303

theory,1 which previously included tasks, norms, tools, and discourse, but did not elaborate the functionof gesturing in the learning and teaching of particular mathematical ideas and concepts in differentialequations.

In the sections that follow, we first review some of the gesture research that has informed our workand we describe how this research fits with a perspective that views mathematics learning as both a socialand an individual process. Next, we delineate the method we developed for documenting and interpretinggesturing in relation to the previously documented evolution of classroom mathematical practices inone differential equations classroom. We then illustrate our analysis of gesturing as it contributes to thedevelopment of several taken-as-shared mathematical ideas for differential equations. Finally, we reflecton our analysis to highlight the contributions of this work and we point to a number of new questions thathave arisen for us as a result of this analysis.

2. Theoretical background

Over the past decade cognitive scientists have begun focusing their research on embodied learning,that is, how a person’s learning is inseparable from her physical being (Lakoff & Johnson, 1999; Lakoff& Nunez, 2000; Varela, Thompson, & Rosch, 1991). This theoretical commitment has, in part, fuelededucational researchers’ interest in gestures and consideration of how gestures relate to learning anddiscourse. Two primary theoretical positions have been developed to describe the relationship betweengestures and discourse. One position is that gestures are simply extensions of language that convey nothingdifferent than words (e.g.,Hadar & Butterworth, 1997). A second position, and the one that informs ourwork, views gestures and language as inseparable, both having semantic value and arising together in theprocess of thinking and/or communication (Cassell, 2000; McNeill, 1992). In order to situate the currentwork within the gesture literature, we next highlight different classificatory schemes for gestures andthe different functions of gestures that have been proposed. We then relate this review to the focus andpurpose of this paper.

Kendon (2000)roughly defined gestures as “the range of visible bodily actions that are, more or less,generally regarded as part of a person’s willing expression” (p. 49). This may include, for example,facial expressions, body language, and hand movements. For our purposes, however, we limit gestures tomovement made by a hand with a specific form: the hand(s) begins at rest, moves away from the positionto create a movement, and then returns to rest (Roth, 2001). McNeill (1992)classified such hand gesturesin terms of the following four types: beat, deitic, iconic, and metaphoric gestures. Beats are motions thathelp emphasize or keep a rhythm in a person’s speech; deictic gestures point or indicate something; iconicgestures imitate some thing by physically looking similar; and metaphoric gestures are like metaphors inthat they have representative meaning, but do not necessarily look like what they are gesturing about.

In another classification scheme,Crowder and Warburton (1995)identified the following three typesof gestures that children and teachers used in an elementary classroom: participatory, outsider, and insider

1 Similar to whatSimon (1995)refers to as a hypothetical learning trajectory, the differential equations instructional theoryincludes, in addition to sequences of tasks, a description of and rationale for a conjectured learning path as it relates to the tasksand the central mathematical concepts and procedures, elaborations on anticipated social and sociomathematical norms thatcan be profitable for promoting the mathematical agenda, and frameworks or other conceptual resources useful for teachers ininterpreting, understanding, and responding to student reasoning.


gestures. In outsider gestures, the person signaling maintains distance between herself and her hands. Aparticipatory gesture involves putting the person as the central point of reference; she stands squarelyinside the space of the gesture. An insider gesture is somewhere between the two, where the self is notexclusively outside or inside and moves back and forth in these metaphorical locations.

In addition to identifying and classifying gestures, researchers have also begun to investigate thefunction of gestures. For example,Clark (1996)proposed a semiotic function of gestures; gestures, aswell as language, tools, and symbols, are signs and these signs are part of a relationship connecting anobject, a sign, and an interpretant (a person for whom the sign means something). Clark suggests that signsexist only as being created by people and for people; this creation of signs is called signaling. Gesturesthen as signs are part of an interactive environment where something can only be a sign if it exists for thesigner and is interpreted as such by the person receiving the sign.Clark (1996)also suggests that thereare three types of signaling, and gesturing can be any of those types; the three types of signaling (signs)include demonstrating a thing (icons), indicating a thing (indices), and describing a thing (symbols).Reynolds and Reeve (2002)found that gestures serve the function of obtaining and holding the attentionof others during their communication and that deictic (pointing) gestures are particularly important tocommunication in small groups when students have trouble explicating their thoughts.

Gestures can also function as indicators for teachers about what is going on for an individual child.For example, when students are at a point that they are ready to make cognitive reorganizations, theirgestures often do not match their words and so teachers might use this mismatch as a clue to informthem about what students have yet to know (Alibali & Goldin-Meadow, 1993). In another study aboutgestures in mathematics,Alibali and DiRusso (1999)illustrated that gestures play an important functionin helping kindergarten students learn to count. Their research indicates that gestures influence a child’sperformance in counting by reducing the mental resources needed as they count because they can usetheir fingers (gestures). As children become more adept at counting, the pointing to keep track drops off,indicating that pointing may actually become more internalized. They further suggest that as a studentgrows mathematically, less and less gesturing may occur as more mathematics is internalized.

Our review of the gesture literature found that the vast majority of studies contribute to a deeperunderstanding of gestures by either outlining different types of gestures (e.g., participatory, outsider, andinsider gestures) and/or by describing the function that gestures play in an individual’s learning (e.g.,reducing the mental resources needed to learn to count) or the function gestures serve between the gesturerand the person attending to the gesture (e.g., when a teacher attends to a gesture-discourse mismatch). Thesignificance of these studies is that they begin to delineate critical features of how individuals constructmeaning and the role of gesturing in this process. That is, previous studies have primarily leveraged ananalysis from an individual point of view. Even when the analysis focuses on the interactions betweenindividuals, say a teacher and a student, the emphasis is on what this implies about the cognition of thoseinvolved. Notably missing from the literature are studies that take a collective lens in examining thefunction of gesturing. The analysis that we include in this paper is one attempt to explore the collectivegesturing activity in one classroom community. To be clearer, we do not take an individual perspectivein this paper when analyzing the gesturing activity of the classroom participants. For instance, to do thismight be to look at whether a student is taking an insider, outsider, or participatory perspective whengesturing. Instead, we take a collective lens to describe the gesturing activity that emerges as classroommathematical practices become taken-as-shared within the community.

In our view, the meanings associated with gesturing activity emerge in interaction among people and areinseparable from discourse. Our theoretical position stems from the symbolic interactionist perspective


which views meaning as mutually constructed in the interactions among participants (Blumer, 1969). Thisis not to deny that gestures have meaning for individuals as well. Rather, the meanings associated withgestures areboth individually and socially constructed. This is consistent with our theoretical positionon learning (the emergent perspective), that learning is both a social and individual process (Cobb,1996). Because we view learning “as both a process of active individual construction and as a processof mathematical enculturation” (Cobb, 1996, p. 35), the theoretical insights gleaned about the meaningof gestures for individuals is important background for analyzing the collective function of gesturing inthe evolution of classroom mathematical practices. A central premise in the emergent perspective is thatthe norms and practices of a classroom do not exist apart from the cognitive reorganizations made byindividuals. Similarly, an individual’s cognitive reorganizations do not exist apart from the evolution ofnorms and practices of the communities in which she participates (Cobb & Yackel, 1996). While we findgreat value in documenting the meaning that gesturing has for individuals, we contend that gesturing isan underdeveloped yet potentially important aspect of the norms and practices that are established withincommunities of learners. We therefore foreground the function of gesturing in collective learning. Assuch, we tell only half of a good story regarding gesturing and mathematical learning.

In addition to committing to a collective theoretical lens, we also want to stress that gesturing is ahuman activity. That is, we will no longer speak of students’gesturesas if they are something apart fromstudents’ reasoning. Rather, we consciously choose to talk of “gesturing” or “ gesture activity” to highlightour position that gesturing is aprocessthat is tied to thinking and reasoning and is not separate fromthe cognizing individual (cf.Rasmussen, Zandieh, King, & Teppo, in press, for a recasting of advancedmathematical thinking to advancing mathematical activity).

3. Methodology

In a previous analysis of the classroom mathematical practices in an introductory course in differentialequations primarily for engineers,Stephan and Rasmussen (2002)documented the emergence of sixmathematical practices over the course of 22 days of instruction on first-order differential equations. Thisanalysis was part of a 15-week classroom teaching experiment (Cobb, 2000) that sought to explore theprospects and possibilities of adapting the instructional design theory of Realistic Mathematics Education(Freudenthal, 1991; Gravemeijer, 1999) and inquiry oriented approaches to university level instructionthat have proven effective for promoting students’ conceptual development at the K-12 level (Rasmussen,Marrongelle, & Keynes, 2003; Yackel, Rasmussen, & King, 2000).

Through systematic analysis of daily classroom videorecordings, this previous analysis of classroommathematical practices usedToulmin’s (1969) model of argumentation as an analytic tool for determiningwhen a particular idea was taken-as-shared. We use the term argumentation in the same manner asKrummheuer (1995)who defined it as a “social phenomenon, when cooperating individuals try to adjusttheir intentions and interpretations by verbally presenting the rationale of their actions” (p. 229). AccordingtoToulmin (1969), an argumentation consists of at least three parts, called the core of an argument: the data,claim (or conclusion), and warrant. In any argumentation the speaker makes a claim and if challenged, canpresent evidence or data to support that claim. The data typically consist of facts that lead to the conclusionthat is made. Even so, a listener may not understand what the particular data presented has to do withthe conclusion that was drawn. In fact, she may challenge the presenter to clarify why the data led to theconclusion. When this type of challenge is made and a presenter clarifies the role of the data in making


her claim, the presenter is providing a warrant. Another type of challenge can be made to an argument.Perhaps the listener understands why the data supports the conclusion but does not agree with the contentof the warrant used. In other words, the authority of the warrant can be challenged and the presenter mustprovide a backing to justify why the warrant, and therefore the core of the argument, is valid.

Stephan and Rasmussen (2002)contend that a mathematical idea that is being discussed as a conclusionbecomes taken-as-shared when either (1) the backings and/or warrants for that conclusion no longerappear in students’ explanations and therefore the mathematical idea in the core of the argument standsas self-evident, or (2) any of the four parts of an argument (data, warrant, claim, backing) shift position(i.e., function) within subsequent arguments and are unchallenged. Using these two criteria we analyzedthe discourse over the first 22 days of instruction by documenting all the conclusions (and associateddata, warrants and backings) that the teacher and students discussed. Then, we traced the function of theteacher’s and students’ contributions over time to see if any of their data, conclusion, warrants and/orbackings shifted function in subsequent argumentations. If either of our two criteria were satisfied, weconcluded that a mathematical idea was taken-as-shared. We then collected all of the taken-as-shared ideasand organized them around consistent mathematical activity, which we labeled as a practice.Table 1showsthe results of our analysis.

Table 1Six classroom mathematical practices

Practice one: predicting individual solution functionsPredictions are exponential in form, andPredictions are based on a rate of change equation that “models” the real world scenario

Practice two: refining and comparing individual predictionsUsing a recursive process yields graphs that are approximations, not exactThe smaller time change increments that are taken in the method, the more accurate the prediction/graphThe exact solution functions use instantaneous rates of change where the approximation graph is comprised of linear stretches

of rates of changeThe graph of the approximation goes up as the time increments are chosen to be smaller (during portions where the solution

is concave up),The initial slope at time zero is tangent to the exact solution, andThe initial slope is the same no matter if the approximation is over a 1-year time period or a half-year time period.

Practice three: creating and structuring a slope field as it relates to predictingReasoning about the way in which slopes change as solutions approach equilibriumSlopes are invariant horizontally for autonomous differential equations, andInfinitely many slopes are encountered in a slope field but only finitely many are visible.

Practice four: reasoning aboutP as both a variable and a functionP can be interpreted as both an unknown function and as an explicit variable in a rate of change equation.

Practice five: creating and organizing collections of solution functionsThe graphs of solution functions do not touch or cross each other (at least for equations studied thus far)Two graphs of solution functions are horizontal shifts of each other for autonomous differential equationsSolution functions can be organized with different inscriptions, andThe phase line signifies the result of structuring a space of solution functions.

Practice six: reasoning with spaces of solution functionsReasoning with dy/dt vs.y graphs across various scenarios,Reasoning dynamically with dy/dt vs.y graphs to structure the general phase line for the bifurcation diagram, andConnecting the dots of several phase lines to result in a “general phase line” (a continuous image of structured solution spaces

that underlie bifurcation graphs).


Fig. 1. Overlapping classroom mathematical practices.

The six mathematical practices documented in this classroom each involved from one to six taken-as-shared ideas (shown as bullets in the table). Because a number of these taken-as-shared ideas are referredto later in this paper, we list all the ideas associated with each practice inTable 1.

In addition to contributing to an emerging body of research on student learning in social context, thisprior work added two theoretical advances regarding the nature of classroom mathematical practices.Both of these advances point to how classroom mathematical practices are different from a delineationof the scope and sequence of mathematical content to be covered in a course. The first theoretical ad-vance involved the notion that classroom mathematical practices can be established in a non-sequentialtime fashion. For example, on the second day of class an aspect of the third and an aspect of the fifthmathematical practice (i.e., slopes are invariant horizontally for autonomous differential equations andtwo graphs of solution functions are horizontal shifts of each other for autonomous differential equations,respectively) were first discussed (but we argued they were not taken-as-shared until later in the course)while aspects of the first two mathematical practices were in the process of becoming taken-as-shared. Toillustrate this complexity, we offer the diagram inFig. 1as an expression of the overlapping characteristicof classroom mathematical practices. The positioning of the boxes is intended to illustrate when theseparticular ideas emerged as a part of an argumentation, with TAS standing for the day that an idea becametaken-as-shared.

The second theoretical advance is that classroom mathematical practices can emerge in a non-sequentialstructure. By this we mean that, on some occasions, we characterized some taken-as-shared ideas asfunctioning in the constitution of more than one practice. This is different than overlap in the timing ofwhen the practices are initiated and constituted as described in the first theoretical advance. For instance,on several occasions students argued for why it should be the case that the slope field for an autonomousdifferential equation should have slopes that are invariant across time, which was a taken-as-shared idea inclassroom mathematical practice three. Students’ arguments regarding invariance also contributed to theconstitution of classroom math practice four, for example. The point that we drew from this is that taken-as-shared ideas do not always function in the constitution of only one classroom mathematical practice– they may contribute to the emergence of other practices and form a network of practices instead of asequential chain of practices with distinct taken-as-shared ideas.


3.1. Coordinating mathematical practice analysis with gesturing analysis

As previously stated, the purpose of this paper is to illustrate an approach for coordinating analyses ofthe collective development of mathematical meaning with the function of gesturing. Before delineatingthe systematic steps we took to do this, we first set forth a number of orienting questions that guidedour analysis. First, in what ways does the second of the previously described two theoretical advancesabout mathematical practices have a gesturing counterpart? More specifically, do we see parallels and/ordiscrepancies in the data that the same gesturings are functioning in the constitution of more than one prac-tice? Second, what is the significance of instances when particular gesturings no longer appear as ideasbecome taken-as-shared? Third, in what ways do particular gesturings shift function within argumenta-tions? For example, in the analysis done byStephan and Rasmussen (2002), an utterance that initiallyfunctioned as a conclusion in some instances shifted to function as data in a subsequent argumentation,evidencing that a particular idea was taken-as-shared. Do we see a parallel for how gesturing functions asideas become taken-as-shared? These last two questions are, to a modest degree, gesturing counterpartsto the two previously outlined criteria used to determine that an idea had become taken-as-shared.

3.2. Method

We began our analysis by systematically reviewing videorecordings from two cameras (one at the frontof the room focused on students and the other at the back of the room focused on whomever was at theboard) from the first 11 days of class.2 During small group work, each camera focused on a set group ofthree to four students. Using the criteria for deciding when a hand movement is or is not a gesture outlinedby Kendon (2000)andRoth (2001), we carefully reviewed the videorecordings and noted all student andteacher gesturings.3 In order to organize our observations and to serve subsequent analysis, we developeda table for each day of class. The various columns in this table identified the person gesturing, the time thegesturing occurred, whether the gesturing occurred in small group work or in whole class discussion (andwhich camera), a descriptive label for each gesturing, the discourse (if any) accompanying the gesturing,and a column for comments, conjectures, and ideas to follow up on.4 During the 11 days, this fine-grainedanalysis noted over 300 gesturings, with 42 different types of gesturing. A sample entry from this table(which we call the Level One Table) is shown inFig. 2.

After generating this first table and realizing the tremendous number of gestures5 (which we hadnot anticipated) we then developed a new, two-column table (what we refer to as a Level Two Table)

2 In the previous analysis of classroom mathematical practices, Stephan and Rasmussen analyzed the first 22 days of class.Since by the 11th day of class a number of ideas had become taken-as-shared and because on the 12th day significantly newmathematical ideas and inscriptions were developed, we chose to focus on first 11 days of class. Moreover, by the 11th day wehad already noted over 300 gesturings and so our decision to stop after the 11th day was also pragmatically oriented.

3 On a few occasions we did not note a gesture if, in the opinion of all three authors, it did not pertain to a mathematical idearelated to differential equations. We also did not include beat gestures.

4 Although we created this table by anticipating what aspects of classroom activity would be most useful for us to categorize,we note the similarity of our table to the one created byReeve and Reynolds (2000). Were we to redo the analysis, we wouldadd a column for inscriptions.

5 While analyzing the raw data, we initially wrote the word “gestures” as a label without drawing out explicitly that wewere analyzing gesturing. To be consistent with our theoretical perspective, however, we use the term gesturing throughout theremainder of the paper except when we refer to the slope shifting gesture as it was originally labeled in the table, leaving thenotion that gestures embody human activity implicit.


Fig. 2. An example from the Level One gesturing table.

Fig. 3. An example from the Level Two Table.

to further organize the gesturings with an eye toward clarifying and illuminating gesturing patterns.The first column of this table was a list of all the different types of gestures. We then developed asecond column for each day that enumerated each gesture, indicated whether this was a teacher orstudent gesture, and whether the gesturing occurred in one of the two small groups being videotapedor if it occurred in a whole class discussion.6 We used black to indicate gestures that occurred inwhole class, blue for gestures that occurred during seatwork in the front group on camera, and redfor gestures that occurred during seatwork in the back group. For example, if the teacher used a dy-namic single slope hand gesture in whole class and this was the sixth gesture of the day, we wrotedown T6 in black (see Level Two Table inFig. 3). If a student in the front group gestured during seat-work and this was the 20th gesture of the day, we wrote down S20 in blue (seeFig. 3). Since the useof different colors is not feasible in this paper,Fig. 3 indicates gesturing in the front group with thesubscript F.

Finally, we coordinated the complete version ofFig. 1 with the Level Two Table into a single ta-ble (called a Level Three Table, seeAppendix A), which illuminated prominent gesturings associatedwith the evolution and use of taken-as-shared ideas. For example, looking at Level Three Table inAppendix A, we noticed that on Day 5 four ideas arose in conversation while two types of gestur-ing appeared (i.e., one hand slope and shifting slope with point). We then used the Level Three Tableto search for patterns and to develop conjectures, going back to the previous tables and videorecord-ings when necessary to clarify events, gather additional information as needed, and confirm, refute andin some instances form new conjectures. To facilitate this process, we grouped the 42 different types

6 When documenting the initiation and constitution of classroom mathematical practices, we look mainly at the conversationsthat take place in the public discourse (i.e., whole class discussions) where everyone theoretically has access to the ideas beingexpressed. However, when we documented the gesturings, we decided to code whether the gesturing appeared in whole classdiscussion versus small group work so that we could use our table for future analyses that investigate other ideas besides collectivelearning.


Fig. 4. Examples of slope shifting gestures.

of gestures into eight different clusters of related gesture activity. We determined gesturing clustersbased on our interpretation of their semiotic function (Clark, 1996). For example, we labeled one clus-ter Slope Shifting Gesture because although there were some idiosyncratic differences in the form thegesture took (e.g., pointed finger, one open hand, two open hands, etc.) the communicative and semi-otic value was similar. Examples of what Slope Shifting Gestures typically looked like are shown inFig. 4.

In the next section of the paper we illustrate our analysis of one of the gesture clusters focusing onshifting slopes and/or graphs of solutions to autonomous differential equations.

4. Coordinating gesturing and math practice analyses

Over the first 22 days of the classroom teaching experiment, six mathematical practices emerged,involving anywhere from one to six taken-as-shared mathematical ideas each as detailed inTable 1.The process by which these practices became established was quite complex in that the mathematical


ideas emerged in a network-like manner, not linearly as previously found byStephan (1998). Inother words, there was not always a marked separation between practices since the instructionalsequence encouraged students to build a web of ideas that integrated their reasoning with graphical,analytic, and numerical techniques. Another notable feature of the classroom mathematical practiceanalysis conducted byStephan and Rasmussen (2002)was the careful attention to both discursiveactivity (classroom talk) and symbolizing (e.g., drawing diagrams, writing equations, generatinginscriptions). As previously noted, however, the analysis did not take gesturing into considera-tion.

In expanding the analysis to include gesturing, we take the perspective that learning is distributedacross discursive activity, symbolizing, and gesturing. Gesturing and symbolizing are semantically tiedto discursive activity in that their meanings become taken-as-shared in argumentation. Furthermore,students’ gesturings are semiotically tied to the inscriptions and tools they create and use. Often gesturingsare created when referring to inscriptions and one can therefore trace the semiotic history of students’gesturings to the meanings created while symbolizing. For the purpose of this paper, we choose tofocus explicitly on two aspects of the triad – discursive activity and gesturing and treat symbolizingimplicitly. The classroom mathematical practice inStephan and Rasmussen (2002)focused explicitly onthe symbolizing and argumentation aspects; hence we will investigate the gesture/argumentation pair indetail for this analysis. We define a gesture/argumentation dyad (or pair) to be an instance of classroomtalk in which a mathematical idea is expressed along with a gesturing. We speak of these two aspects as adyad to be consistent with our theoretical commitment to treat argumentation and gesturing as inseparableactivities.

The results of our analysis suggested to us that gesturing/argumentation dyads changed function acrossinstruction, i.e., when students’ arguments shifted from CONCLUSION7 to DATA, the function of thegesturing associated with the argumentation changed concurrently. Therefore, we found that certaingesture activity supported the emergence of classroom mathematical practices in the following two ways:

1) A gesture/argumentation dyad can emerge during the establishment of one or more of the taken-as-shared ideaswithin the same practice. For example, we found that a particular gesture/argumentationdyad first emerged during the establishment and re-establishment of the second taken-as-shared ideaunder classroom mathematical practice three. Then, later the same dyad was used to draw a newCONCLUSION, that there are infinitely many slopes in the slope field that solution curves encounter(the third idea in classroom mathematical practice three).

2) A gesture/argumentation dyad that emerges while establishing one practice can change function tosupport the establishment of ideasacross other practices. For example, the same dyad that supportedthe development of ideas within classroom mathematical practice three was used to support the es-tablishment of ideas within classroom mathematical practices two and five.

In the next section we provide examples to illustrate these two ways that gesturing can support the de-velopment of classroom mathematical practices, while emphasizing the dyadic relation between gesturingand discursive activity.

7 Each time we discuss one of the components of Toulmin’s model of argumentation, such as data or conclusion, we writethe word in all capital letters to explicitly highlight that the contribution served that communicative function.


Fig. 5. Graphs of the changes in a population of rabbits for two different starting scenarios.

4.1. Gesturings that support the establishment of one or more of the taken-as-shared ideas within thesame practice

One of the mathematical ideas that became taken-as-shared during the development of the third class-room mathematical practice was the notion that slopes are invariant horizontally for autonomous differ-ential equations (seeTable 1andFig. 1). This idea first emerged as a topic of conversation briefly on thesecond day of the instructional sequence. The task on that day was to draw a graph to show how a popu-lation of rabbits might change over time if there were unlimited resources and continuous reproduction.As students discussed their drawings the following figure was drawn on the chalkboard (seeFig. 5). Thefigure shows the anticipated growth of the rabbit population starting with either an initial population of10 or 20 rabbits.

One student CONCLUDED that the beginning slope on the solution curve (for population of 20)would be the same as the slope on the 10-curve, just at a later time. Although this conclusion wasnotated by the teacher with tangent vectors as shown inFig. 5, neither other students nor the teacherelaborated on this CONCLUSION and no gesturing acts were made by any of the participants. The ideaof horizontal invariance in slopes was resurrected, however, as a topic of conversation on the fourth dayof instruction when the teacher gave students a partially filled out slope field for the equation dP/dt =3P(1 − P/100) and asked them to fill in the missing slopes and to explore patterns in the grid of slopesand implications of these patterns for graphs of solutions. As students investigated these patterns theystarted concluding that the slopes for a starting population, sayP = 10, would be the same no matter thetime.

For example, in one small group Marla argued that time was not relevant to the initial slope.

Marla: If the population at sixty [Marissa: is sixty initially] then it can be sixty at time zero or itcan be sixty at time seven. It would not matter.

Marissa: [quietly] Sixty at time zero or sixty at time seven.

Frank: So the seven would be zero because that would be your starting point.

Marla: Exactly.

Marissa: Say what you just said again.

Frank: Seven would be zero because that would be your starting point.

Marissa: So if we just shifted the whole [SLOPE SHIFTING GESTURE – seeFig. 6]

(. . .)


Fig. 6. Marissa’s slope shifting gesture.

Marla: If we’re at time seven, then we can call seven our initial, you know, this is our initial time.At time seven, if our initial population is sixty, then what is the rate of change?

Marissa: Oh, it would be the same as it is right here.

Marla: Exactly. And at time four, the population is thirty. Then what would be the rate of change?

Marissa: Whatever it is at thirty.

Marla: Exactly. So you’ll notice that the rates of change all the way across here [SLOPE SHIFTINGGESTURE] are identical. Time is not a factor [SLOPE SHIFTING GESTURE] as far asthe rate of change [goes].

As illustrated inFig. 6, the slope shifting gesture refers to a left–right translation using one or twohands, usually taking the iconic form of a slope mark. In noting when gesturing occurred in relation towhat was said, we indicate the gesture activity in brackets immediately following the words that werespoken concurrently with the gesture activity. We follow this convention in the remainder of this paper.

Here we see an example of student-initiated gesturing during small group work that accompanied aCONCLUSION that the slopes were the same all the way across the time axis. As it turned out, theother small group on camera was also debating the same CONCLUSION but their argumentations didnot include gesturing. The idea of slope invariance became a topic of conversation during whole classdiscussion as well, and although neither the teacher nor the students accompanied their arguments witha slope shifting gesture, the mathematical idea of slope invariance with respect to time for autonomousequations became taken-as-shared in this class period (seeStephan & Rasmussen, 2002, for our evidencefor this).

Students again discussed the idea of invariance of slopes, the following class period, but using thedifferential equation as DATA for their CONCLUSION instead of supporting the CONCLUSION byreasoning about patterns in the slope field, as was the case on the previous day. Therefore, the CONCLU-SION that slopes are invariant across time arose again and was re-negotiated (i.e., was further supportedby new evidence, the rate of change equation). It was during this conversation, using the rate of changeequation as DATA to support the fact that the slopes are invariant across time, that the slope shifting


gesture appeared for the first time in whole class discussion. As Julie pointed out about the differentialequation, “Nothing is depending on time. It’s like dependent on the population. It makes sense with theslopes being all the way across at a certain population [SLOPE SHIFTING GESTURE].” The teacherfollowed up her contribution by saying, “You’re saying it’s all the way across like this?” and pointed to theslope field. Later during the same class period, the teacher used the slope shifting gesture to summarizestudents’ CONCLUSIONS that the slopes are the same for a given population whether you start at time0 or time 2:

T: We say at time two we are starting with one-tenth rabbits. So, the number of rabbits we are startingwith is one-tenth. So, the rate of change that we would then move forward from at this point is stillthree-tenths, right? We still have this same slope here, in other words [SLOPE SHIFTING GESTUREwhile pointing to (P = .1, t = 0) and to (P = .1, t = 2)]. Or if we started at time two with five-tenthsrabbits, we still have this one right here [SLOPE SHIFTING GESTURE while pointing to slopemarks at (P = .5, t = 0) and (P = .5, t = 2)]. Or if we start at time 2 with one rabbit, we would stillhave that same slope, that rate of change, slope, right there [SLOPE SHIFTING GESTURE]. Is thatkind of the idea?

In summary, the slope shifting gesture first emerged from students during small group as they tried tocome to grips with whether or not the slopes were invariant across time for autonomous differential equa-tions. Although the invariance of slopes idea became taken-as-shared without the public display8 of theslope shifting gesture, the slope shifting gesture appeared publicly as students refined their conclusion byusing the rate of change equation as further evidence that slopes are invariant for autonomous differentialequations. Therefore, this stands as an example of a gesturing activity that emerged during the establish-ment of a mathematically significant idea within a classroom mathematical practice. Furthermore, in thisexample students’ CONCLUSION of invariance of slopes and associated gesturing (the slope shiftinggesture) emerged as a gesture/argumentation dyad. In the next example we see this gesture/argumentationdyad shift function in the discourse by changing from a previously debated CONCLUSION to serve asDATA for a new CONCLUSION.

The slope shifting gesture appeared again as the teacher and students were drawing CONCLUSIONSabout a different mathematically significant issue, the idea that the solution curves encounter infinitelymany slopes in a slope field (the third idea in classroom math practice three). For example, the teacherused a hand-held calculator to graph the slope field for dP/dt = 3P(1 − P/100) and asked students toexplore how the solution functions would flow within the field and to make sense of the graphs of exactsolution functions as compared to graphs of approximations. Below, Marissa describes to the whole classone of the things her group noticed:

Marissa: We also saw that you could go to the population of thirty and we just chose the time intervalof zero to one, but there’s infinitely many of the, rate of change is the same. There’s infinitenumber of those hash marks in between zero and one that you could make your, just drawthem in.

T: Yeah, when I chose this, when I made this slope field to make slopes at zero, one, two, and three,her point is that you could have chosen to make slope marks at zero, one-tenth, two-tenths,three-tenths, etc.

8 By public display, we mean those gestures or arguments that were made public in whole class discussion.


Marissa: And at thirty it’s the same all the way across [teacher and Marissa simultaneously display aSLOPE SHIFTING GESTURE].

T: It’s the same all the way across [SLOPE SHIFTING GESTURE]. You bet. So, one of thethings you might imagine yourself doing as you’re using this slope field is that you start withthis rate of change at fifteen of twenty-four and six-tenths and then you imagine yourself goingup, but then you’re increasing the rate of change [draws on the slope field to show a smoothcurve drawn in instead of line segments]. Notice that my movement there was not dictated bythe numerical approximation but by a sense of the rate of changes I’d be encountering as Iwent.

In this example, Marissa and the teacher related the notion that the slopes were invariant across timeto the idea that there were infinitely many of those invariant slopes in any time interval for a givenpopulation and that the solution functions were encountering them as time goes by. It was as though thenotion that the slopes are invariant across time was given in this discussion (i.e., it was taken-as-shared)and Marissa and the teacher described how the solution curves interacted with these slopes in the field.The teacher capitalized on Marissa’s contribution and even repeated the slope shifting gesture as here-explained her “encountering slope” idea.

This example shows that gesture/argumentation dyads that originate during discussions of one mathe-matically significant idea can be used to draw new CONCLUSIONS (e.g., that solution functions are en-countering infinitely many slopes in a slope field). We have thus far discussed how gesture/argumentationdyads can emerge in students’ argumentations during the establishment of one or more of the taken-as-shared ideas within the same practice. Next, we turn to the other way in which gesturing can support thedevelopment of classroom mathematical practices, that dyads originating during the development of oneclassroom mathematical practice can re-emerge to support the emergence of other practices.

4.2. Gesturings that support the establishment of taken-as-shared ideas within other practices

In the previous examples, we showed that gesture/argumentation dyads that originate as studentsestablish a certain mathematical idea that can shift function and be used to establish other ideas within thesame classroom mathematical practice. As we will show in the classroom episodes that follow, the samegesture/argumentation involving the slope shifting gesture supported the emergence of mathematicalideas in two different classroom mathematical practices. One instance in which this occurred was onthe eighth day of the instructional sequence in which students were trying to articulate the relationshipbetween the graph of an exact solution and the graph of an approximation, both of which were shown ona slope field generated by a hand-held calculator. By making this comparison, the notion that the exactsolution is comprised of instantaneous rates of change (part of classroom mathematical practice two)became a topic of conversation and the dyad involving the slope shifting gesture again emerged:

Tom: Well, the instant of time that the graph actually hits, you know, the beginning of each populationmark, where the slope marks start, that’s going to be the slope [of the exact curve]. But, afterthat, it’s going to change.

T: Say it again. I’m not sure I was following you. Just repeat what you said.

Tom: At the instant of time that the graph reaches, say, like the population, like where the slope marksfirst start, then it’s going to be that slope at that instant of time, but after that, it’s going to change.


Fig. 7. Ray’s slope shifting gesture.

T: Wait, wait. So you guys agree with that? At this instant in time where it starts, the exact solutionis going to have the same slope as all the approximations? Right?

A few seconds later, two students display the slope shifting gesture to further support Tom’s argumentabove:

Ray: No, actually, I agree with Tom because you’re readjusting your slope at every interval. So, at theend of the interval, it’s going to be off, it’s going to have a different slope. But once you get tothe beginning of the new interval, it’s going to have the exact same slope as the function [SLOPESHIFTING GESTURE – seeFig. 7].

In this example, Ray used the gesture/argumentation dyad that the slopes are invariant across time as agiven (i.e., he makes the assumption that the other students believe that the slopes are invariant) in order todraw a new CONCLUSION, that the slopes on the approximation are the same as the slopes on the exactsolution at the beginning of the time interval for the slope mark for the approximation. In other words,invariance of slopes was taken-as-shared by this point in the classroom and students, like Ray, were usingthe slope shifting/invariance dyad as evidence to describe the relationship between exact solutions andapproximations. The teacher pushed this notion a bit further by asking the following question:

T: Yeah, so here’s the question. Suppose I’m doing this rough approximation with a deltatat one-tenth, right? So at time zero, we have the exact same slope, rate of change, as thefunction that we know, the e to the three t. And I go to my next time increment, right here,right? And I recalculate a slope. What does this recalculated slope have to do with the slopeof the exact function?

Marissa: There’s a point on the graph of the exact function, where there’s an instant where the slope isexactly the same [SLOPE SHIFTING GESTURE]. When it reaches exactly that population.

T: So, there’s a point where it reaches this one [referring to rough approximation curve] thatthis slope [the exact function curve] has the same one?

Marissa: Right. An instant when each one reaches that specific population, where it’s the same. But,it just gets there a lot quicker.


T: Um, gets there a lot quicker? So, this one [the exact function curve] got here a lot quickerthan this one got to here [referring to the rough approximation curve]. Right?

Marissa: Because there’s no linear increase in the exact function.

In this case, the mathematically significant idea that the exact solution functions are comprisedof instantaneous rates of change that are the same as the approximations along the same populationvalue emerged as a topic of conversation on day eight and became taken-as-shared that day. Thegesture/argumentation dyad of slope shifting gesture and invariance of slopes across time was integralto this establishment. By integral we do not mean a causal relationship, but rather that the dyad servedas DATA. Students like Ray and Marissa used the dyad as DATA to CONCLUDE that the exact andapproximation graphs’ slopes are related in that the exact solution function hits the same slope as theapproximation only at an earlier time. This illustrates how the function of a gesture/argumentation dyadcan shift across classroom mathematical practices. In other words, in earlier class sessions the notionthat slopes are invariant across time for autonomous differential equations was a CONCLUSION thatwas debated by students. Then, the slope shifting gesture emerged from students as they used the rateof change equation to re-establish that slopes are invariant across time. The slope shifting gesture andthe idea that slopes are invariant across time for autonomous differential equations was established as agesture/argumentation dyad that was used in subsequent arguments.

To illustrate the shifting function of the gesture/argumentation dyad further, we turn to an episode thatoccurred on the tenth day of the instructional sequence. In this example, students were working in smallgroups on the following problem:

Two cups of hot coffee are left in a room with an ambient temperature of 72◦F. Assuming thatNewton’s law of cooling applies, (i.e., that the rate of cooling is proportional to the differencebetween the current temperature and the ambient temperature), the rate of change equation dT/dt =−0.2(T − 72) has been proposed as a means for predicting the future temperatures as a function oftime.

1. Suppose one of the cups of coffee is initially 180◦F and other is initially 160◦F. Is one cup of thecoffee cooling off more rapidly that the other at this instant?

2. Will there ever be a time when the two cups of coffee are the exact same temperature? Why or whynot? Try and come up with more than one reason for your answer.

Two different students used the slope shifting gesture/invariance dyad to make their CONCLUSIONSabout whether or not graphs of solution functions would ever touch.

Dick: So what you’re saying is, this is showing that one hundred eighty is just the same as one hundredsixty except the one hundred sixty is just shifted over [SLOPE SHIFTING GESTURE] once youget to a certain period of time. . .? All right.

Paul: One hundred eighty is just one hundred sixty shifted [SLOPE SHIFTING GESTURE] over, howmany time units.

While this shows evidence that students, in small groups, were using the slope shifting ges-ture/invariance dyad as support (i.e., DATA) for the fact that one cup of coffee cools off more rapidly thananother, the same gesturing/argumentation dyad emerged in whole class discussion, first by a studentand then by the teacher. As Josh explained his group’s findings to the class, he used the slope shifting


gesture/invariance dyad to conclude that the rate of change for the 180◦ cup of coffee will never catch upto the 160◦ cup.

Josh: They’ll never catch up because the rate of change is dependent on just the temperature and nottime. So initially if you start off at one hundred eighty degrees, you’re gonna start with a rateof change that’s steeper, but it’s like an exponential decay. Once you get to one hundred sixty,you’re gonna have the same rate of change, the same slope as one hundred sixty just at a latertime [SLOPE SHIFTING GESTURE].

T: So this slope and this slope will be the same just at a shift over [SLOPE SHIFTING GESTURE]?

Josh: Yeah. So technically until both of them finally get to the cap. . . [they will not catch up].

In this example, Josh and the teacher both used the slope shifting gesture/invariance dyad as support(i.e., DATA) to CONCLUDE that the two solution curves will never catch up to each other. This is anotherillustration of the idea that gesture/argumentation dyads can shift function and be used as support formore sophisticated argumentations across different practices. In this case, a gesture/argumentation dyadthat was established in classroom mathematical practice three was used in establishing a mathematicallysignificant idea in classroom mathematical practice five.

As our final example of this phenomenon, consider the episode that occurred on day 11 of the in-structional sequence in which the class revisited the temperature problem to discuss whether or not twosolution curves (one starting at 180◦ and one at 160◦) would touch each other. Paul used the slope shiftgesture/invariance dyad as support to CONCLUDE that the two graphs would not touch.

Paul: I don’t know, I kind of thought about it like this. We know from some of the stuff we’ve alreadydone that the two graphs are the same graph, they’re just shifted you know shifted along the hori-zontal by how many time intervals. So I guess you can kind of think about them as parallel at everypoint, parallel or they have a parallel tangent at every point because those slope lines are the sameacross whenever along the y [SLOPE SHIFTING GESTURE]. So if the graphs are just shiftedover from each other why would they ever touch each other? If they are indeed the same graph?

In this example, Paul used the same gesture/argumentation dyad again as evidence to draw the CON-CLUSION that two graphs will never touch each other. The notion that two solution curves for autonomousdifferential equations will never touch or cross was a mathematically significant idea that became taken-as-shared later in instruction, and as we see from these examples, the class used the gesture/argumentationdyad as evidence to support the establishment of this idea.

5. Conclusion

The preceding analysis sheds new light on gesturing in the mathematics classroom and the functionof gesture/argumentation dyads as they contribute to the evolution of classroom mathematical practices.The theoretical position that we took in this analysis is that gesturing cannot be viewed apart from sym-bolizing and discursive activity. Moreover, mathematical meaning is not located in a gesture, inscription,or the words someone says, but rather, mathematical meaning emerges through human activity. We haveread where some researchers (Alibali & Goldin-Meadow, 1993) say that they turn off the volume of theirvideotape in order that they can focus on the visible form of gestures without interference from the thingspeople say. Although we see value in this approach for certain purposes, this methodological approach


would not be a viable way for us to glean insight into the function of gesturing in argumentation. For us, ar-gumentation and gesturing work as an indivisible unit and therefore we created the gesture/argumentationdyad construct to frame our analysis.

By tracing the function of these gesture/argumentation dyads, we contribute to understanding thefunction of gesturing in two ways. First, we extend the earlier work ofPea (1993)by acknowledging thatknowing is distributed across use of not only tools and inscriptions, but also gestures and discourse. Indoing so, we focused on gesturing as opposed to gestures since we view meaning not in a gesture butrather part of human activity that involves expressing, communicating, and reorganizing one’s thinking.Second, we contribute new insight to the research on social construction of meaning. Using the sociallens of the emergent perspective,Cobb and Yackel (1996)define the classroom mathematical practiceconstruct as taken-as-shared ways of reasoning mathematically, often involving symbolizing. We expandthis construct to include students’ and the teacher’s gesturing activity in the constitution of the collectivelearning of a classroom community. As such, we add to the gesture literature in that little research hasfocused on gesturing as a collective activity in a classroom setting. Many contemporary researchersanalyze gestures from an individual perspective. In contrast, we use a social lens to analyze gesturingsas they emerge in the interactions among classroom participants and contribute to the constitution ofmathematical meaning.

In the beginning of this paper we set forth the following three guiding questions stemming from theearlier analysis byStephan and Rasmussen (2002):

• In what ways do we see gesturings functioning in the constitution of more than one practice?• In what ways does gesturing activity shift function within argumentations?• What is the significance of instances when particular gesturings no longer appear as ideas become

taken-as-shared?

Our analysis provides some answers to the first two questions and the third raises more interestingquestions. Regarding the first question, we gave examples where gesture/argumentation dyads wereestablished in the classroom in small groups and in the whole class discussion. They did not appear ina linear fashion, but created a network of dyads that wove together in the constitution of more than oneclassroom math practice. We pointed to the slope shifting gesture as one of the many types of gesturingsthat were used in the first 11 days of the class. This gesturing appeared as classroom mathematicalpractice three became established, but it also played a role in the renegotiation of that same classroommathematical practice on a later day with different data in the argumentation.

As far as gesturing shifting function within argumentations, our analysis suggested that this did happenwith the slope shifting gesture/invariance dyad. This dyad, which was constituted early on as data toconclude that an infinite number of slopes are encountered in a slope field9 and other classroom mathpractices, shifted its function on day 10 and 11 (and possibly other times as well). At these points,the gesture/argumentation dyad became a warrant used in the negotiation of the taken as shared ideathat an exact solution is comprised of instantaneous rates of change (part of classroom mathematicalpractice 2) and that solutions to autonomous differential equations never touch or cross (part of classroommathematical practice 5).

9 A slope field for a differential equation dy/dt = f(t, y) is a sketch in the (t, y) plane of minitangent lines whose slope isf(t,y) at selected points. A slope field offers a graphical means to visualize graphs of solutions to the given differential equationwithout a priori knowing the analytic form of the solution functions.


Fig. 8. Sample table outlining aspects of a measurement instructional sequence.

Regarding the third question, we did not see certain forms of gesturing drop off as particular ideaswere constituted. We note, however, that we did not analyze any argumentations in whichStephanand Rasmussen (2002)found students’ contributions to fall off in subsequent argumentations. We arestill interested in this question and in subsequent analyses we will look explicitly at these argumentationinstances to see if particular gesturings drop off. As an alternative to dropping off, we might find thatcertain gesture/argumentation dyads “slide under” other gesture/argumentation dyads, thus creating achain of dyads, each building on the one that is under it (cf.Walkerdine’s, 1988notion of a chain ofsignification). We may also find that a gesturing actually re-forms with different mathematical ideasforming new gesture/argumentation dyads. Finally, some dyads might eventually join together to formmore complex dyads that contribute to the constitution of increasingly sophisticated mathematical ideas.

Along with these inquiries about our guiding questions, we conclude with some other ideas that areworth investigating. For instance, how might a gesturing analysis inform instructional design efforts? Inthe instructional design theory of Realistic Mathematics Education, some researchers are delineating theuse of tools, imagery, activity, and discourse as part of describing the intention of instructional sequences.For example,Gravemeijer, Bowers, and Stephan (2004)organize these four aspects of a measurementinstructional sequence in a table that connects their function. A sample of this table is shown inFig. 8.Our analysis on the function of gesturing in mathematical learning suggests that gesturings could be auseful addition in such a description. The table would then show particular gesturings that might occurspontaneously from students as practices are established and that classroom teachers might draw onin discussions. Additionally, the table could suggest gesturings that could be introduced by the teacherherself to support certain mathematical ideas.

Lastly, we worked from a previously determined set of classroom mathematical practices. Now thatgesturing is explicitly part of our awareness, how might an initial analysis for practices proceed if it weretaken into consideration from the start? Are there additional aspects to practices that we “missed” in theearlier analysis? That is, might a practice analysis that includes a focus on gesturing illuminate additionalor different taken-as-shared ways of reasoning or underlying imagery? In what ways might a teacher findinformation about gesturing as it relates to the evolution of particular mathematical ideas useful in herday-to-day practice? We look forward to investigating further some of these questions.

Acknowledgements

Support for this paper was funded in part by the National Science Foundation under grant No. REC-9875388. The opinions expressed do not necessarily reflect the views of the foundation. The authorswould like to thank Randi Engle for her comments on an earlier draft of this paper.

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Appendix A

Sample from the Level Three Table


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