Using the K5 Connected Cognition Diagram to analyze teachers’ communication and understanding of regions in three-dimensional space

Ucs

DU

a

A

KKMSSSSTT

1

bowrc

imkgakblec

0d

Journal of Mathematical Behavior 31 (2012) 235– 251

Contents lists available at SciVerse ScienceDirect

The Journal of Mathematical Behavior

journa l h o me pag e: ww w.elsev ier .com/ locate / jmathb

sing the K5 Connected Cognition Diagram to analyze teachers’ommunication and understanding of regions in three-dimensionalpace

eborah Moore-Russo ∗, Janine M. Vigliettiniversity at Buffalo, The State University of New York, Department of Learning and Instruction, 566 Baldy Hall, Buffalo, NY 14260-1000, United States

r t i c l e i n f o

vailable online 29 December 2011

eywords:5 Connected Cognition Diagramultimodality

emiotic bundlesemiotic connectionsemiotic resourcesemioticseachers’ subject matter knowledgehree dimensions

a b s t r a c t

This paper reports on a study that introduces and applies the K5 Connected Cognition Diagramas a lens to explore video data showing teachers’ interactions related to the partitioning ofregions by axes in a three-dimensional geometric space. The study considers “semiotic bun-dles” (Arzarello, 2006), introduces “semiotic connections,” and discusses the fundamentalrole each plays in developing individual understanding and communication with peers.While all teachers solved the problem posed, many failed to make or verbalize connectionsbetween the types of semiotic resources introduced during their discussions.

© 2011 Elsevier Inc. All rights reserved.

. Introduction

Graphing plays a fundamental role in mathematics. The application of the Cartesian coordinate system to plot relationsetween input and output variables is a mainstay of mathematics at the secondary and postsecondary levels. National Councilf Teachers of Mathematics (2000, 2006) has recommended graphing in the upper elementary grades to study patterns asell as in the high school years to “investigate conjectures and solve problems involving two- and three-dimensional objects

epresented with Cartesian coordinates” (NCTM, 2000, p. 308). This study considers teachers’ understanding of the Cartesianoordinate system.

Effective teachers have general pedagogical knowledge that helps them create classroom environments that foster mean-ng making, but they also rely on more than general pedagogical strategies that work in any subject area. Effective teachers

ust understand the specific content they are teaching. Shulman (1986) introduced the domains of pedagogical contentnowledge and content knowledge refocusing the efforts of the educational research community to consider more thaneneral pedagogical knowledge. Pedagogical content knowledge refers to an understanding of the subject matter, students,nd curriculum at such a level that a teacher knows how to best represent and sequence topics to maximize learning. Contentnowledge, in the context of mathematics education, refers to an understanding of mathematical objects, the relationshipsetween mathematical objects, and the mathematical knowledge of the conventional principles (e.g., order of operations,

aws of exponents) that guide mathematical activity. Ball, Thames, and Phelps (2008) expanded on Shulman’s content knowl-dge by including the following three subdomains: common content knowledge, specialized content knowledge, and horizonontent knowledge.

∗ Corresponding author. Tel.: +1 716 645 4069; fax: +1 716 645 3161.E-mail addresses: [email protected] (D. Moore-Russo), [email protected] (J.M. Viglietti).

732-3123/$ – see front matter © 2011 Elsevier Inc. All rights reserved.oi:10.1016/j.jmathb.2011.12.001

dx.doi.org/10.1016/j.jmathb.2011.12.001

http://www.sciencedirect.com/science/journal/07323123

http://www.elsevier.com/locate/jmathb

mailto:[email protected]

mailto:[email protected]

dx.doi.org/10.1016/j.jmathb.2011.12.001

236 D. Moore-Russo, J.M. Viglietti / Journal of Mathematical Behavior 31 (2012) 235– 251

Common content knowledge includes the basic mathematical knowledge and skills that are essential to and used in a widevariety of settings. With respect to graphing, teachers’ common content knowledge allows them to recognize that the point(2,3) is situated in the first quadrant two units in the positive direction on the x-axis and three units in the positive directionon the y-axis. Teachers’ content knowledge should extend past common content knowledge to a more robust understandingof both the mathematical concepts themselves as well as the ways students visualize, communicate and reason about theseconcepts.

Teachers’ specialized content knowledge is an understanding of mathematics that is both flexible and nuanced. As Balland Bass (2000) point out, it is this knowledge that allows teachers to “mediate students’ ideas, make choices about rep-resentations of content, modify curriculum materials, and the like” (p. 97). With respect to graphing, teachers’ specializedcontent knowledge allows them to (1) recognize that ordered pairs written in the form (x, y) and points plotted on a Carte-sian coordinate system are both common representations of the same mathematical object, (2) move fluidly between theserepresentations choosing the most appropriate for the situation at hand, and (3) realize that the horizontal positioning ofthe x-axis and the vertical positioning of the y-axis are a matter of convention rather than a requirement.

The final subdomain, horizon content knowledge, involves understanding how mathematical knowledge is related andconnected to more advanced mathematical concepts. With respect to graphing, working in two dimensions logically extendsto working in three. Although the formation of the octants in three dimensions is not traditionally a concept taught in highschool mathematics, it is one that high school mathematics teachers should have experienced in their university calculuscourse sequence. Teachers’ horizon content knowledge allows them to recognize how the formation of the quadrants in twodimensions segues into the formation of the octants in three-dimensional space via orthogonal axes. Teachers should beable to facilitate development of students’ notions related to the two-dimensional Cartesian coordinate system in such away that will allow the students to build on this understanding should they continue on to courses that would involve thegraphing of functions with two independent variables.

This study proposes a multimodal conceptual framework to consider teachers’ subject matter knowledge. Specifically,the study uses teachers’ interactions to a series of tasks related to the number of regions formed by the x-, y-, and z-axesin the three dimensional Cartesian coordinate system. The consideration of teachers’ communicating and reasoning aboutmathematical concepts, especially those related to basic three-dimensional objects, requires a multimodal lens. Recognizingthe rich context and multiple semiotic resources that characterize human interaction and thought, this paper draws onVygotskian perspectives of learning and the work of others related to semiotics in mathematics education. The goal of thiswork is twofold: to consider how teachers communicate and reason about basic three-dimensional objects and to show thatthe proposed conceptual framework is of value when analyzing teachers’ communication involving mathematical concepts.

2. Conceptual underpinnings

Semiotics is the study of signs (things that represent or convey information); how they are used to signify actions, ideas,and objects; and how these signs are interpreted1 (Hoopes, 1991; Radford, 2002). We will refer to signs as semiotic resources,as is preferred by those in social semiotics, to “avoid the impression that ‘what a sign stands for’ is somehow pre-given, andnot affected by its use” (Van Leeuwen, 2005, p. 3).

A semiotic set, also referred to as a register by Duval (2006), is comprised of semiotic resources of the same type (e.g., a setof words as opposed to a set of symbols). For example, the algebraic register includes coefficients, variables, and exponentsas elements. Another example of the semiotic set commonly used for graphing includes the horizontal and vertical markingsthat represent each of the axes as well as the small dots that stand for points. A semiotic system is comprised of threecomponents: a semiotic set, the mode of production for the elements of the set, and the relationships between the elementsof the set (Arzarello, 2006). Different semiotic systems are associated with various mathematical forms.2 The semioticelements used for equations and the relationships between them are very different than the semiotic elements used forother mathematical forms such as tables, graphs, or diagrams.

Arzarello (2006) introduced the semiotic bundle, which is comprised of different semiotic sets and the relationshipsbetween them. Often different types of semiotic resources naturally occur together in pairs, as is seen in the speech–gesturedyad when saying the word “yes” while nodding one’s head. Another example in which the types of resources are comple-mentary would be the gesture–artifact dyad of tracing one’s finger on a map to signify a route.

A semiotic bundle is a dynamic structure, changing as the semiotic systems employed by the individual or group developand evolve. Duval (2006) distinguished between two categories of this evolution. He considered transformations from onesemiotic representation to another where both either (1) evoked the same mathematical form (treatment) or (2) referenced
different mathematical forms (conversion). For example, writing a series of equivalent, yet distinct, algebraic equations (e.g.,−2x + y = 4 and y = 2x + 4) is a treatment. In contrast, a conversion occurs in a transformation from an algebraic equation likey = 2x + 4 to its graph on the Cartesian coordinate system. In both examples, the mode of production for the semiotic sets
1 Charles Saunders Peirce first introduced the triadic relation of signs.2 In keeping with the language of Duval (2006), we use the term “mathematical form” as a reference to the form of the mathematical process rather

than using the term “mathematical representation” as is employed around discussions of the Rule of Four (i.e., the use of equations, graphs, tables, verbalscenarios).

uws

nhtbtiep(

mtsHiawwr2

KDabK

teTauttr

2Aruftaftfw

scafbd

te

D. Moore-Russo, J.M. Viglietti / Journal of Mathematical Behavior 31 (2012) 235– 251 237

sed is the same (marks on a paper). In the treatment example, the transformation occurs between equivalent equationshich both involve elements of the same semiotic set. However, the conversion example involves moving from one semiotic

ystem (with symbols used to represent equations) to another system (with the symbols used to represent graphs).Traditionally, cognitive activity has been considered an internal process. In this view semiotic resources externalize inter-

al processes. However, recent work (Núnez, Edwards, & Matos, 1999; Radford, Bardini, Sabena, Diallo, & Simbagoye, 2005)as suggested that semiotic resources are more than simply tools of communication. Individuals develop understandinghrough the interplay of multiple types of semiotic resources (Radford, 2008), and cognitive activity is embedded in theody’s experiences (Barsalou, Niedenthal, Barbey, & Ruppert, 2003; Lakoff & Johnson, 1980; Smith, 2005). What happens inhe mind is dependent not only on the task at hand, but also on what is happening with the body, how the body is situatedn and interacts with the environment, and the peculiarities of the environment in which the body is situated. The idea ofmbodied cognition blurs the distinction between the mind and body (Lakoff & Johnson, 1999) by proposing that individuals’erceptions and interactions guide and are guided by the semiotic resources they use and those in their environments useThom & Roth, 2011).

Semiotic resources are socially embedded and culturally created (Greeno, 1997); the manner in which they mediateathematical activity depends on the particular task at hand as well as the previous experiences of and interaction between

he individuals engaged in the task. Semiotic resources are created, selected, and given meaning, often unintentionally andubconsciously, by the people using them in their attempts to understand and share mathematical ideas (Moreno-Armella,egedus, & Kaput, 2008). Semiotic resources not only influence how a person comes to understand a concept, they also

mpact the use of other semiotic resources. For example, the presence or absence of modeling materials can impact the typend number of gestures a person uses when dealing with a concept. A person’s use of certain gestures can in turn impact theords he selects when talking about the concept. Moreover, the words spoken regarding a concept can in turn influence theritten markings (including the algebraic symbols, graphs, drawings, etc.) that are used. It has been suggested that semiotic

esources not only affect reasoning, they can also determine the contents of reasoning as well (Alibali, Phillips, & Fischer,009; Goldin-Meadow & Beilock, 2010).

This paper introduces and applies a conceptual framework which arose from postsecondary instructional experiences, the5 Connected Cognition Diagram. The K5 Diagram also serves as a visual representation that extends the works of Arzarello anduval, using four types of semiotic resources to highlight the fundamental role each plays in an individual’s understandingnd communication with others. Data from classrooms were codified and graph theory was employed to establish relationsetween the use of semiotic resources. We aim to show that the data provided here support the notion that the use of the5 Diagram assists conceptual framing.

While one might intuitively grasp the significance of the term “connected” from its use in common vernacular, it helpso be familiar with mathematical terminology to fully appreciate the name of the K5 Connected Cognition Diagram. A math-matical graph contains vertices and edges. A complete graph is one whose edges directly connect all vertices in the graph.he graph theory notation for a complete graph on five vertices is K5. A connected graph is a graph for which there exists

path (not necessarily direct) from any vertex to any other vertex on the graph. While it is mathematically redundant tose both complete graph notation and to also include the word “connected” in the diagram’s label, this purposed repe-ition highlights the idea that although direct links between types of semiotic resources are possible, it is often the casehat a person negotiates the path from one semiotic resource to another via a third, or even a fourth type of semioticesource.

There are multiple ways that a single concept can be conceptualized, represented, visualized, or communicated (Duval,008). The K5 Connected Cognition Diagram emphasizes a discrete distinction between four broad types of semiotic resources.dmittedly this diagram oversimplifies the dynamic nature of interacting semiotic sets within a given type of semioticesource (such as the relationship between algebraic and graphical written markings). However, it is helpful in trying tonpack the complex interplay of other transformations between semiotic resources. Conversions involve the use of trans-ormations from one semiotic representation to another where each references a different mathematical form. However,ransformations from one type of semiotic resource to another where the mathematical form remains unchanged shouldlso be considered. For example, a teacher could graph a parabola on the board and then use his index finger and thumb toorm a U-shape that represents the same parabola. The referent (the graph of the parabola) remains constant; however, theypes of semiotic resources used vary. For this paper, we introduce the term semiotic connection to mean any transformationrom a semiotic representation used to refer to a concept or object to another which represents the same concept or object,here the semiotic system used changes, the form of the referent changes, or both change in the transformation.

The K5 Diagram features the four types of semiotic resources determined by their modes of production. Each of theemiotic modes serves as a medium for thinking-through-articulating by allowing individuals a means to represent andommunicate mathematical ideas. The meaning is not held in the semiotic resources themselves but in how they are usednd interpreted. Each type of semiotic resource has particular affordances and drawbacks in terms of the mathematicalorms it is able to represent. The K5 Diagram also features the semiotic connections, including conversions, which are madeetween and within these four types of semiotic resources. Next we describe the four types of semiotic resources in more
etail.
The first type of semiotic resource, inscriptions, includes written words, abbreviations, markings, shapes, or drawingshat by association or convention are used to represent something. For example, common mathematical inscriptions includequations, tables, graphs, geometric figures, and Venn diagrams.


Inscriptions Artifacts

Speech Gestures

Fig. 1. K5 Connected Cognition Diagram (top view).

The second type of semiotic resource, speech, involves (but is not limited to) the spoken word. Tone of voice as well asverbal utterances such as tongue clicks, sighs, onomatopoeia, and nonutterances (such as pauses in speech) are also includedin this type of semiotic resource. For example, when discussing the x-intercepts of a parabola, using the phrase “the graphdecreases then, BOOM, it is at the x-axis” exemplifies how speech including onomatopoeia can be used in mathematics as asemiotic resource.

Gestures, the third type of semiotic resource, involve placement or movement of a part (or multiple parts) of the body todraw the attention of another to an object, to emphasize a point made in speech, to send a message in place of speech, or torepresent an object or scenario through embodiment. While gesturing can be intentional, it is often done in a spontaneous,unconscious manner. Italianate and beat gestures (McNeill, 1992) as well as facial expressions are included in this categorybut are not considered in this work as they were of less interest than mathematical gestures in the current research. Bymathematical gestures, we mean gestures used to represent mathematical ideas during communication with self and others.Forming a parabola with the hand, pointing to a drawing of a parabola, or tracing a parabola in the air with a finger are threecommon mathematical gestures. For more information on gesture, Goldin-Meadow and Beilock (2010) provide a succinctoverview of this type of semiotic resource.

The fourth and final type of semiotic resource, artifacts, includes physical objects, electronic devices or digital tools used asa means to accomplish a task. Pencils, rulers, calculators, and virtual manipulatives are common examples of mathematicalartifacts. Artifacts can be classified as media, tools, or instruments, which all serve to represent information. For example, onpaper (medium) a person uses a pen (tool) to draw a set of axes (instrument). Note that an artifact is not always implementedin its prescribed manner. Holding two pencils together to demonstrate two intersecting lines would be an instance in whichartifacts are used as instruments to complete a task for which they were not originally created (see Fig. 1).

A mental representation is the “way the world is internalized by the individual” (Radford, 1998, p. 291) regarding a par-ticular concept in a particular setting at a particular moment in time. Mental representations are framed by and frame thesemiotic resources a person uses. They are shaped by the context in which they arise as well as the cultural and personal his-tory (the lived experiences and perceptions) of the person producing them. While we agree with Lesh, Post, and Behr (1987)that the “external/internal dichotomy” between semiotic resources and mental representations is artificial, the differencein mental representations and the other four types of semiotic resources is that the four are observable. Therefore, we focuson the four types of observable semiotic resources that not only provide a window to an individual’s mental representationbut also have the power to alter existing and create new mental representations.

Changes in a person’s mental representation of a concept impact and are impacted by how concepts are communicated,visualized, and reasoned. Through a process of coevolution, the inscriptions, speech, gestures, and artifacts used influenceand are influenced by how a person thinks about a concept. This is best seen from a different angle of the K5 Diagram (seeFig. 2). It is through this mutual interaction between mental representations, inscriptions, speech, gestures, and artifacts
that a person comes to grasp a concept. Moreover, people who “understand” a concept should be able to call on a varietyof semiotic resources choosing the most appropriate for the situation and task. To return to the diagram, individuals withstrong, fluid connections between all five vertices on the K5 Diagram understand more deeply than those who are missingconnections or for whom the connections are weak and not easily made.
Speech Gestures

Artifacts

Mental Representation

Inscriptions

Fig. 2. K5 Connected Cognition Diagram (side view).

ifftsc

uirrba

3

3

afclTiap

gods

sTeauao

gTBoobi

ig

3

sawa


Besides the conceptual framing previously mentioned (emphasizing semiotics and the K5 Diagram), the research was alsonspired by Sfard’s and Kieran’s (2001) description of their own mid-research paradigm shift in which their focus changedrom thinking as a regulator of communication to communication as an act of meaning making. In a similar manner this studyocused on small groups’ collaboration as the unit of analysis, their “activity of communication” (p. 72), purposely blurringhe line between individual and social research perspectives. Whereas the cognitivist and the interactional approacheseparate knowledge construction to internal and external processes, this study adopts the stance that they are both ways ofonsidering what is a single phenomenon.

We now turn to educational episodes using the four types of observable semiotic resources in the K5 Diagram to helpntangle the complex interactions of adults collaboratively engaged in mathematics. This research was driven by the follow-

ng questions. Which of the four observable semiotic resources found in the K5 Diagram do teachers use to communicate andeason about regions in three-dimensional space? What does the K5 Diagram reveal about how teachers communicate andeason about regions in three-dimensional space? What does the K5 Diagram reveal about how the teachers use connectionsetween semiotic resources to make meaning while engaging in mathematical tasks? What does the K5 Diagram revealbout the connections made between semiotic resources?

. Methodology

.1. Participants and setting

Video and written data were gathered during three sections of the same mathematics education graduate course over period of 2 years at a research university in the northeastern United States. All three sections of the course had identicalormats. Each class met on a weekly basis and was divided into five groups, each with four to five members. Groups wereomposed of a mixture of preservice and inservice teachers. The inservice teachers were all New York State certified, hadess than 3 years teaching experience, and held undergraduate degrees in either Mathematics or Mathematics Education.he preservice teachers all had either a Bachelor’s degree in Mathematics or held a Bachelor’s degree with at least 30 creditsn undergraduate mathematics courses at the level of calculus or beyond. The participants of the study will be referred tolternately as group members or teachers. For the purposes of this paper, the term teachers will be used to refer to bothreservice and inservice teachers.

The instructional design of the course fostered teacher–teacher interaction allowing extended involvement in smallroup work as a means to construct, negotiate, verify, and refine the teachers’ mathematical ideas. For example, one seriesf activities involved the following. First, individuals created definitions for slope that would work in both two and threeimensions, then they worked together as a small group on a single collective definition for slope. For more details on thiseries of activities see Moore-Russo, Conner, and Rugg (2011).

By philosophical design, instructor interventions were limited and brief. The instructor circulated repeatedly among themall groups of teachers observing each for brief intervals rarely making any attempt to refocus or redirect discussion.he instructor usually answered teachers’ questions with additional queries intended to help the group revisit, discuss, andventually answer their own questions. Each small group was video recorded, allowing the instructor access to all discussionsfter class. In terms of the meta-discursive rules, the classroom environment seemed to invite both proactive and reactivetterances during group discussions. Video records provide evidence that the teachers in the small groups were initiatingnd responding to each other in a manner that suggested they were interested in the interaction and sensitive to the needsf their group members.

This study focuses on related episodes from two different sections of the course described above that involved two smallroups of teachers who worked on the same series of problems. Group One consisted of two inservice teachers (Carl andina) and three preservice teachers (Bonnie, Jaime, and Kendra). Tina scored in the bottom 5% on the spatial literacy pretest;onnie, Jaime, and Kendra scored in the middle while Carl scored in the top 5%. They were sitting in a circle in the followingrder: Carl, Tina, Jaime, Bonnie, and Kendra (who was also next to Carl). Neither Bonnie nor Kendra minded if their handsr bodies were recorded, but both preferred not to have their faces on camera. For that reason, the camera was positionedetween and behind them. Due to this positioning, at one point Kendra leaned back and was completely off camera (and

mpossible to code) for approximately 2 min during the episode.Group Two consisted of one inservice teacher (Kara) and three preservice teachers (Dave, Donna, and Ruth). Kara scored

n the bottom 5% on the spatial literacy pretest; Donna and Ruth scored in the middle while Dave scored in the top 5%. Thisroup sat in a circle in the following order Ruth, Donna, Kara, then Dave (who also sat next to Ruth).

.2. Educational materials

Both episodes described below came from the second recorded class session. Prior to the episode under consideration the
mall groups engaged in collaborative work on a series of tasks related to the Cartesian coordinate system, points, distance,bsolute value and midpoints in one- and two-dimensional environments. As in the first recorded class session, each groupas provided with a single copy of text. The teachers were instructed to work through the text with its embedded questions
s a group, but they were required to provide written solutions individually that were submitted at the conclusion of each


S

N

EW

above sea level

below sea level

Fig. 3. First figure in course text.

class session. The text for the class session under review began by using the two-dimensional environment to introducethree-dimensional space. A section of the text read:

Let the x-axis represent east/west movement, the y-axis north/south movement, and add an additional axis, one that isperpendicular to the others, to define a three-dimensional system of coordinates. Let’s call that third axis the z-axis; thinkof it as being above or below sea level.

A rendering of a standard, three-dimensional Cartesian system, which made use of geographical orientations (as seen inFig. 3), immediately followed this text.

Next the text introduced ordered triples and the plotting of points in three dimensions. A second figure with the sameorientation as the first was provided but with x-, y-, and z-axis labels in place of the W–E, S–N, and below–above sea levellabels, respectively as seen in Fig. 4. This figure also showed how to plot the ordered triple (−7, 4, 6).

The first embedded question in the text asked for the teachers to write the coordinates for the ordered triple that rep-resented the origin, which had been introduced as the point where the x-, y-, and z-axes meet. The second question thatimmediately followed asked:

If the axes of the xy-plane divide it into four regions (called quadrants), how many regions do the axes of the xyz-spacedivide it into? Explain your reasoning.

At the beginning of the course, each teacher was provided with a kit whose primary purpose was to model three-dimensional (3D) situations (McGee, Moore-Russo, Ebersole, & Lomen, in press). The 3D kit uses a system of stacking pegsto place magnets in a 3D environment (3-space). These magnets are then strategically placed to hold up representations ofpoints, curves, vectors and surfaces in 3-space.

3.3. Data collection

Prior to instruction an instructor-created pretest was administered to gauge the spatial literacy of the teachers in thegraduate course. The spatial literacy pretest was administered to the 64 teachers enrolled in the three sections of thecourse over 2 years. Items on the pretest covered topics in two and three dimensions such as basic geometric objects (e.g.,points, lines, planes, circle, triangles), relationships between geometric objects (e.g., rotations, intersections, bisections and
parallelism), and applications of measurement to geometric objects in 3-space.
Video data were collected for each 160-min class session. Cameras were set up prior to each class to reduce distractions.These cameras recorded all group work for the full length of the class. In addition to the video data, teachers’ individuallysubmitted written work was also reviewed.

Fig. 4. Second figure in course text.

rwG

3

lht(ot

adpat

ernw

tfirKnc

c1w

rostcr

4

tsot

4

tt(


Both groups spent approximately 15 min completing the series of tasks. Focusing on the time spent on the questionegarding the number of regions in 3-space, the clock began after the group completed reading the question and endedhen all group members finished their written solutions. Group One worked on this particular question for 10 min 2 s whileroup Two spent 11 min 25 s on this question.

.4. Data analysis

The video data under analysis were selected for three reasons. First, the research team deemed the series of tasks high-ighted in the episodes below as being integral to understanding 3-space. Second, each of the two groups had a member whoad scored in the bottom 5% on the spatial literacy pretest. Not only did the data provide an opportunity to consider howhe groups approached the tasks in general, it also provided an opportunity for the research team to juxtapose the actionsand interactions) of the teachers with weaker spatial literacy. Third, the interactions of the members, including their usef semiotic resources, in the two groups were identified as being particularly interesting in comparison with each other forhis series of tasks.

As Barwell (2009) suggested data are not transparent. Two researchers could observe the same situation yet providelternate accounts due to differing theoretical perspectives. To minimize subjectivity, the research team discussed all analysisecisions until consensus was reached using collaborative microgenetic techniques (Schoenfeld, Smith, & Arcavi, 1993)rimarily based on the analysis of the teachers’ videotaped interactions as noted by their use of inscriptions, speech, gestures,nd artifacts. In addition, the research team also used the written data from the teachers’ collaborative work to ensure thathe essence of the episodes was captured.

During the initial stage of analysis, both members of the research team viewed the video data together. First, each grouppisode was observed in its entirety as the groups completed the series of tasks. Next, each problem-solving episode wasepeatedly watched, often in slow motion, and narrative descriptions of each video were created. Upon completion of eacharrative, the work was reread alongside the video it aimed to capture. This process was repeated until the research teamas satisfied that the narratives were an accurate portrayal of the video data.

Working independently the members of the research team coded each narrative according to the semiotic resources ofhe K5 Diagram. The agreement for narrative coding was 0.99 with Cohen’s kappa 0.97. Both team members discussed theirndings until consensus was reached for all coding. For ease of reference coding that involved multiple types of semioticesources was included in the narrative descriptions beginning with inscriptions (I) and rotating counterclockwise about the5 Diagram to include speech (S), gesture (G), and artifacts (A). If a particular event included both speech and gesture, thearrative was coded (SG). Some interactions were elaborate requiring several sentences of explanation. In these cases theoding was given at the conclusion of the description.

After consensus was reached, the two members of the research team revisited the videos while reading the narratives toonfirm their final coding and to determine the counts for each code. Next the research team reviewed each of the possible5 coding categories (I, S, G, A, IS, IG, IA, SG, SA, GA, ISG, ISA, IGA, SGA, and ISGA) to compile summaries of each. The categoriesere based on the combinations of the four types of semiotic resources: inscriptions, speech, gesture, and artifacts.

After assigning codes to the teachers’ activity, the data were reviewed for instances of treatments (transformations ofepresentations within a semiotic set where the referent remained constant) and semiotic connections (transformationsf representations used to denote the same concept or object but where the form of the mathematical referent or theemiotic set used for the referent changed). The research team reviewed the coding categories assigned to each teacher inhe episodes. Those instances where one teacher or multiple teachers used semiotic resources in succession forming semiotichains (Presmeg, 2006) were reviewed using the K5 Diagram until consensus was reached as to whether the activity wasepresentative of a treatment or a semiotic connection.

. Teachers’ uses of semiotic systems and semiotic bundles

The K5 Diagram lends itself to consideration of 15 possible combinations of types of semiotic resources. A single vertex onhe K5 Diagram represents a variety of semiotic systems each with the same mode of production. Four of the vertices representemiotic systems. The remaining eleven combinations represent semiotic bundles comprised of at least two different typesf semiotic resources. To address the first research question, we now consider the semiotic resources used by the teacherso communicate and reason about basic three-dimensional objects by describing and giving examples of each.

.1. Monads

We will first consider the four vertices, or monads, from the K5 Diagram. Teachers in both groups frequently used a singleype of semiotic resource. Examples of inscriptions (I) included teachers considering the figure in the text, redrawing thehree axes with labels to suggest a different orientation than was provided, writing out possible ordered triples (e.g., (+,+,+),+,+,−),. . .), and looking at others’ written solutions.


Examples of speech3 (S) included teachers stating or restating the problem; voicing confusion; making conjectures aboutthe problem or how to proceed to solve the problem; questioning their group members about terminology, orientation,and labeling; and agreeing (or disagreeing) with their group members. At times speech involved metaphorical references.Bonnie referred to the negative z-axis as “digging down in the dirt,” while Kendra referred to the positive z-axis as going upa ladder. Kendra compared planes intersecting in 3-space to cutting a block of JELL-O®. Only Jaime, in Group One, did notuse speech by itself.

For the most part, gesture (G), whether used alone or in concert with another type of semiotic resource, fell into twocategories. In the first, gesture was used to recreate or augment a representation of the intersecting axes. In these cases theteachers used their hands or arms to model the situation in such a way that their bodies became part of the graph. In thesecond category, gestures were indicatory in nature. The teachers’ hands, arms, or other parts of the body were used as ameans to draw attention to an artifact or inscription. In these situations pointing was used as a means to draw attentionto objects rather than represent them. For this reason, the research team used a different code (GP) to denote all pointinggestures that were used to highlight real or imagined objects. Gesture rarely occurred alone. The first instance was whenDave shook his head in agreement, while the other two occurrences involved Tina contemplating the problem on her own bymimicking the gestures used by others. As with gestures, few instances of artifacts (A) occurred. All cases occurred in GroupOne when teachers considered the corner of the room to model the intersection of the three axes as Carl had introduced.

4.2. Dyads

We will now consider the six vertex dyads from the K5 Diagram, in other words instances when two types of semioticresources were used, beginning with dyads that involved inscriptions. Besides teachers reading aloud from the text to theirgroup members, examples of the inscription–speech (IS) dyad involved discussion and questions that focused on the figurein the text, their own drawings, others’ written solutions, and formulas. The following excerpts from the narratives compiledby the research team are representative of the ways in which the IS dyad occurred.

Carl [in Group One] introduced a new inscription as he drew the intersection of only the positive x-, y-, and z-axes extendingfrom the origin on a sheet of paper describing them as “the corner of a room” (IS).

Once Kara [in Group Two] finished reading, she displayed the second figure in the text to the group (I). She paused thencommented that she did not like the position of the z-axis in the figure (IS). . . Dave indicated that orientation of the axesdid not make any difference in the situation (IS), and Ruth verbalized her agreement that it did not matter how the figurein the text was oriented (IS).

The sole instance of an inscription–gesture (IG) dyad involved Carl writing out 2 × 2 × 2 in the air. The inscription–artifact(IA) dyad was not present in the interactions of either group.

Having discussed the dyads that involve inscriptions (including the IS dyad), we now move to the dyads that involvespeech. Of these, IS dyads were most common followed in order by speech–gesture (SG) and speech–artifact (SA) dyads. TheSG dyad involved using gesture and speech to create an imagined set of axes, add the z-axis to a plane, demonstrate how aplane divides 3-space, rotate imagined axes, and consider regions formed by axes. Carl in Group One and Donna in GroupTwo used this dyad the most frequently. The following excerpts are representative examples of how the SG dyad occurred.Each excerpt describes in detail the exact events of the interactions and the order in which they occur.

Kendra [in Group One] explained to the group that she thought of the xy-plane as four quadrants, gesturing by holding bothher hands in front of her in a flat, palm down position to represent a horizontal plane. In unison she moved her hands downand then up, while keeping both palms down, to indicate that there would be four regions where z was negative and fourmore where z was positive (SG). Carl explained, “It’s like a box” drawing out the boxes’ edges with his fingers and indicatingthe eight regions that would be formed using a pinching gesture to signify each region. Each of his drawing gestures wasaccompanied by a certain sound effect while each of his pinching gestures was accompanied by a different sound effect (SG).Kendra compared this to cutting a block of JELL-O®, saying that there would be eight different “squares” (S). Carl agreedand then expanded on this metaphor by using an imagined block of JELL-O® in front of him and slicing it in half with aplane gesture using onomatopoeia. He stated that there would be two sections of JELL-O® following this first slice. Repeatingthe cutting motion with sound effects he said that there would be four sections following the second cut. With one morerepetition he explained that the four sections would become eight (SG).

The speech–artifact (SA) dyad only occurred in Group 2; every instance, with one exception, involved Kara. Kara discussedhow to use the 3D kit to model the three intersecting axes, to clarify axis labels, or to consider the axes’ orientation. The
exception occurred when Dave responded to one of Kara’s queries regarding the 3D kit.
Having discussed the dyads that involve inscriptions and speech (including the IG and SG dyads), we now move to theremaining dyad that involved gesture. The gesture–artifact (GA) dyad occurred just once in each group. In one instance Tina

3 To be included in the speech monad, all other types of semiotic resources, including the use of accompanying gesturing or gazing to draw the attentionof another to an object, were absent.

cr

4

dttaitca

tt

Swt

4

tb


ontemplated the problem using the corner of the room and hand gestures; the other instance involved Dave indicatingegions on the 3D kit as a means to complement Donna’s use of speech.

.3. Triads

We will now consider the four triads from the K5 Diagram, most of which included pointing gestures. Similar to the ISyad, Group Two’s inscription–speech–gesture (ISG) triads involved discussion and questions that focused on the figure inhe text, the teachers’ own drawings, and others’ written solutions. The most common gesturing in the ISG triad involvedeachers’ pointing to inscriptions to draw another’s attention to them. Sometimes this was intended to direct attention ton inscription in general, at other times it was meant to direct attention to a specific part of an inscription. The ISG triadncluded non-pointing gestures in only two instances. The first involved Carl, in Group One, asking questions as he triedo make connections between the orientation of the axes in the text’s figure and a gesture made by Bonnie in which sheonfigured three fingers of one hand (thumb, index and middle), each orthogonal to the other. The second involved Davend is shown below along with other examples of the ISG triad that involved pointing.

Donna [in Group Two] pointed at the figure in the text and indicated the order in which the axes were labeled (ISGP). At thesame time Dave stated, as he drew an imaginary vertical axis with his hand, that he had no difficulties with the orientationof the x- and y-axes as long as the z-axis was vertical (ISG).

Kara [in Group Two] said that she had always struggled with two-dimensional drawings as she pointed to Donna’s drawing(ISGP). Ruth also pointed to Donna’s drawing and agreed that it was difficult to visualize the situation using a two-dimensional drawing (ISGP). Donna began to label the regions on her drawing aloud (IS) as Ruth and Kara glanced ather work (I). Kara asked if Donna had labeled a region correctly in her drawing (IS). Ruth pointed at Donna’s drawing andstarted explaining the directionality of one of the axes to Kara (ISGP).

The inscription–speech–artifact (ISA) triad occurred just once when Carl, in Group One, compared the figure in the texto a drawing he had made. The inscription–gesture–artifact (IGA) triad occurred only once as Kara, in Group Two, pointedo the 3D kit to keep track of the regions while she wrote out her solution on her own.

The speech–gesture–artifact (SGA) triad occurred twice in Group One but 15 times in Group Two. The examples of theGA triad in Group Two all involved the 3D kit. Teachers in this group often pointed to specific regions or parts of the kithen asking questions or when giving explanations to another regarding the partitioning of the regions or the labeling of

he axes. The following excerpts exemplify how the SGA triad was evidenced in both groups.

Carl [in Group One], used his gaze and finger pointing, to indicate where the walls met and said, “in the corner of the room,z is going up” (SGPA). While doing this, all the members of the group turned to look at the corner of the room (A). Carl thenpointed to the wooden chair rail, which ran three feet above the floor on the wall, indicating that the chair rail on one wallwould be the x-axis and the chair rail on the other wall would be the y-axis (SGPA). Tina used her pen lining it up in hersights with the seam of the wall where Carl indicated the z-axis would be (A). . . Then Tina verbally confirmed which chairrail represented the x-axis with Carl, mimicking his pointing gestures (SGPA).

Kara [in Group Two] revoiced her previous statement that the three axes would create four “large” regions as she pointedto each of the four regions where z values were positive on the model (SGPA). This elicited responses of disagreement fromDave and Donna (S). Donna picked up the model Dave had created with the 3D kit and pointed to the z-axis stating that itwas positive above the xy-plane. She then used her finger to become the z-axis below the xy-plane indicating that this waswhere the z-axis was negative (SGA). She pointed to each quadrant of the plane as she counted out loud, indicating with avertical sweep of her hand that those four regions were above the xy-plane and hence z was positive in this space (SGPA).Interrupting her Dave took two pieces of paper and extended the x- and y- axes on the model into orthogonal vertical planesstating, “These are planes.” As he held the partitions in place he and Donna jointly explained that there would be as Davestated, “four regions on top and four below.” (SGA). Kara pointed at the xy-plane and asked (SGPA), “Is this a plane?” Daveconfirmed (S), “Yeah, it’s the xy-plane.” Donna reoriented the plane vertically and used her hand to embody the regionswhere the x- and y-values were positive and negative (SGA). She then repositioned the model so that the z-axis was onceagain vertical (SGA).

.4. Tetrad

The inscription–speech–gesture–artifact (ISGA) tetrad was only present in Group Two. It was seen most often wheneachers pointed to the 3D kit during their interactions to compare it with either the figure in the text or a drawing createdy another teacher. The following excerpts exemplify the ISGA tetrad.

Holding her pen in the middle and moving it side-to-side in a sliding motion to represent the x-axis in the text’s figure, Ruth[in Group Two] switched its position indicating with a hand movement that the x-axis would switch places with the y-axis“coming out” from the figure (ISGA). . . Ruth then relabeled her axes according to the orientation used by the figure in thetext and confirmed this new orientation by pointing first to the 3D model then to the text figure (ISGPA).


Table 1Teacher’s use of semiotic resources from K5 Diagram.

I S G A IS IG IA SG SA GA ISG ISA IGA SGA ISGA

Group OneCarl 5 4 0 0 5 1 0 5 0 0 1 1 0 1 0Tina 5 2 2 3 6 0 0 2 0 1 0 0 0 1 0Bonnie 4 4 0 1 4 0 0 0 0 0 0 0 0 0 0Jaime 2 0 0 1 6 0 0 2 0 0 0 0 0 0 0Kendra 3 2 0 1 1 0 0 2 0 0 0 0 0 0 0

Group TwoDave 6 5 1 0 3 0 0 1 1 1 3 0 0 5 0
Kara 4 5 0 0 10 0 0 0 4 0 2 0 1 3 5Donna 9 4 0 0 11 0 0 4 0 0 5 0 0 4 1Ruth 8 3 0 0 9 0 0 0 0 0 5 0 0 3 2
In order to ensure that her list of possible ordered triples was complete, Kara [in Group Two] pointed to two of the regionson Dave’s model as she discussed the signs of each of the axes with Ruth (ISGPA).

4.5. Summary of teacher’s use of semiotic resources

The above scenarios provided insight into the teachers’ interactions. In this section the frequency with which the teachersused the semiotic systems and bundles is discussed. The frequency of use for each monad semiotic system, dyad, triad, andtetrad semiotic bundles for the teachers in both Group One and Group Two are displayed in Table 1.

Carl introduced multiple representations in an impromptu manner using the four semiotic resource types seen in theK5 Diagram. Throughout the episode, Carl was able to provide multiple representations of the intersecting axes though heoften used representations that included only part of the situation and not all 8 regions. He used more semiotic systems andbundles than the other teachers in Group One and was more likely than them to incorporate gestures and artifacts. The datain this study support the spatial literacy pretest results that placed his spatial literacy in the top 5% of all students who hadtaken the course. Despite this, Carl rarely used the terminology associated with the mathematical objects and properties herepresented.

Tina used almost as many semiotic systems and bundles as Carl, but the way in which she used them was drasticallydifferent. She frequently voiced her confusion. Her use of speech was often to ask questions about the semiotic resourcesused. With statements like “I’m confused” and “I don’t get it,” Tina’s requests for help failed to specify her reasons forconfusion or to identify that she needed help making connections between different representations. The data from thisstudy support the spatial literacy pretest results that placed her in the bottom 5% of all students.

Bonnie, Jaime, and Kendra used the fewest semiotic systems and bundles. We have the least evidence regarding Bonnie’sunderstanding and communication. Even though her posture and gaze suggested that she attended to and understood thegroup’s discussion, Bonnie seemed satisfied with the idea of using ordered triples to explain the situation and never madeuse of other semiotic resources once the triples were suggested. The data suggest that Jaime and Kendra both understoodthe situation. Both used gestures to model the situation, and Kendra in particular provided a written explanation as to whyit was appropriate to use ordered triples to address the problem posed.

In Group Two, evidence of Dave’s understanding of 3-space was demonstrated during the episode through his use ofmultiple semiotic resources. Dave used the 3D kit to model the situation; at one point he extended the axes of the modelon the 3D kit with convenient artifacts (sheets of paper). He also introduced a combinatorial approach relating it to signsof the axes in 3-space. Dave followed the discussions of his group members regardless of the semiotic resource being usedand interceded when he deemed it necessary. For example, Dave did not require a particular orientation for the axes, andon more than one occasion, he was able to reorient the axes to address others’ questions when he gestured, referencedthe model, or referenced two-dimensional drawings. Moreover, Dave consistently labeled his gestures and artifacts usingmathematical terminology to reference both objects (e.g., the xy-plane) and their properties (e.g., the positive x-axis). Thedata in this study support the spatial literacy pretest results that placed his spatial literacy in the top 5% of all students whohad taken the course.

During the episode Kara used a number of semiotic systems and bundles (second only to Donna) and was the person withthe highest number of semiotic bundles involving an artifact, always the 3D kit. Kara’s preference for the use of an artifactto represent the situation was immediately apparent. The number and essence of Kara’s numerous questions, especially atthe beginning of the episode, marked her lack of understanding of the situation and supported the spatial literacy pretestfindings that placed her in the bottom 5% of students. By the end of the episode she was able to use the four semiotic resourcesof the K5 Diagram in harmony to facilitate meaning making and to come to a more complete understanding of the situation.

Donna employed more semiotic systems and bundles than any of the other teachers in Group Two. She spent a large
portion of her time considering ordered triples using her inscription-based, two-dimensional representation of 3-space.Donna’s attachment to the two-dimensional drawing was puzzling as she was the group member who led the discussionwith the model on the 3D kit showing a skeptical Kara that there were eight regions. Even though Donna recognized thatthere were eight regions and demonstrated that she could identify them on the 3D model, she claimed preference for her


tD

bsh

tra

4

mm

4

cO

woiil

4

cT

tamf

Fig. 5. Group One’s use of monads, dyads, triads, and tetrads.

wo-dimensional drawing when creating the list of ordered triples. In the end she was unable to complete the list withoutave’s help.

During the episode Ruth used a number of semiotic systems and bundles involving inscriptions. She discussed the com-inatorial approach with Dave, the 2D drawing with Donna, and the model on the 3D kit with Kara. Although she stated thathe preferred not to work with 2D drawings, she was easily able to do so and was the only person to include a drawing iner submitted work. Data suggest that Ruth understood the axes’ partitioning of 3-space.

A couple of brief notes end this section related to semiotic resources. None of the members of either group used languagehat referred to the orthogonal nature of the axes in either two or three dimensions. Also, neither group referred to theegions as octants, even though the members of Group Two did discuss that is was inappropriate to refer to the eight regionss quadrants.

.6. Application of the K5 Diagram to groups’ uses of semiotic resources

In this section the graph theory roots of the K5 Diagram are applied to the frequencies seen in Table 1 and are used toake evident and analyze the similarities and differences between the interactions of the members of Group One and theembers of Group Two.

.6.1. Group OneThe frequencies for the monads, dyads, triads, and tetrads for the teachers in Group One can be seen in Fig. 5. The five

omponents of Fig. 5 show the cumulative frequency of the four semiotic systems and the eleven semiotic bundles for Groupne.

Group One’s interactions involved numerous instances of the I and S monads as well as the IS and SG dyads. The IS dyadas the semiotic bundle used most often by members of Group One. Not only did the members of Group One make use

f relatively few semiotic systems and bundles (besides I, S, IS and SG), the semiotic bundles that they employed typicallyncluded at most two types of semiotic resources. Despite, the many visually oriented representations of 3-space that werentroduced, Group One eventually agreed to individually write and submit solutions that were strictly inscription-based (aisting of all possible ordered triples in terms of the signs of the x, y, and z-coordinates).

.6.2. Group TwoThe frequencies for the monads, dyads, triads, and tetrads for the teachers in Group Two can be seen in Fig. 6. The five

omponents of Fig. 6 show the cumulative frequency of the four semiotic systems and the eleven semiotic bundles for Groupwo.

Group Two utilized each of the four types of semiotic resources outlined in the K5 Diagram. Even though they spent almost
he same amount of time on the problem as Group One, members of Group Two used almost twice as many semiotic bundlesnd tended to use semiotic bundles that involved more types of semiotic resources. As in Group One, speech was evident inost of the semiotic bundles employed. While the general trends in using the I and S monads as well as the IS and SG dyads
requently is similar in Group One and Group Two, Group Two also made use of the SA dyad, the ISA triad, the SGA triad,

Fig. 6. Group Two’s use of monads, dyads, triads, and tetrads.


and the ISGA tetrad. All four members of Group Two used artifacts (most often the 3D kit) and gesture extensively. Each ofthe group members frequently made reference to the 3D kit gesturing as they identified the number of regions in 3-space.

As did Group One, Group Two came to the consensus that their final written solutions should utilize the ordered tripleinscription-based listing of the eight possible regions. Different than Group One, each of the solutions offered by the membersof Group Two also included a written explanation of why the ordered triples applied to the situation. The conclusion of theepisode found members of Group Two gesturing using the 3D kit to deliberately connect their artifact-tool bundle to thesymbolic representation of the coordinates they were writing on their papers.

5. Teachers’ uses of connections between semiotic resources

This section considers the semiotic connections used by the members of each group. Rather than provide an exhaustivelist, as seen in Section 4.1, of all the connections from each monad, dyad, triad, and tetrad, the K5 Diagram was used to considerthree types of semiotic connections that occurred: (1) conversions which featured transformations from one mathematicalform to another involving different semiotic sets with the same mode of production (shown as one vertex of the K5 Diagrammapping to itself), (2) conversions which featured two different types of semiotic resources (movement from one vertex ofthe K5 Diagram to another) for different mathematical forms, and (3) transformations which featured two different typesof semiotic resources (movement from one vertex of the K5 Diagram to another) for the same mathematical form. Anexample of a conversion that began and ended with semiotic resources from the inscription vertex occurred when Donnamade a conversion from a hand-drawn image of the three axes to an inscription using ordered triples. Bonnie’s conversionfrom the inscription in the text (a two-dimensional drawing of the axes) to a metaphor using speech is an example ofa conversion featuring two different types of semiotic resources. Many instances of non-conversion, semiotic connections(i.e., transformations between types of semiotic resources with the same mathematical referent) occurred as students movedfrom inscriptions that represented the intersection of the three axes and to gestures that represented this intersection.

While considering the data, the research team often found that a semiotic bundle featured one particular type of semioticresource although multiple resources were present. In order to simplify the language used in analysis, the research teamdenoted semiotic connections by the type of semiotic resource featured at the end of the connection. For example, eachgroup reader used an IS dyad to introduce the problem to the group (speech that related the figure in text). The focus ofthis interaction was on the inscription rather than the speech. After the IS dyad was used to introduce the members ofGroup One to the problem, Bonnie immediately used speech, a metaphor, to represent the situation. This was counted as aconnection from an IS dyad that featured an inscription (the diagram in the text) to an S monad. To simplify, this examplewas referred to as a connection featuring speech, since that was the featured semiotic resource at the end of the conversion.The following example demonstrates how Carl used a semiotic connection from a semiotic bundle that featured gesture toa semiotic bundle that featured an artifact (his pen representing the z-axis), then made a final connection to yet anothersemiotic bundle that featured a different artifact (the corner of the room) where all bundles featured the same referent. Bothsemiotic connections were labeled as featuring artifacts.

Carl used the following dynamic gesture with one of his hands to help Tina visualize the situation. Using a flat hand, palm-sidedown in a raised position to represent the xy-plane, he allowed his hand to drop with a dramatic sound effect to emphasizethat the xy-plane “fell” onto the desk in front of him stating “think of the xy-axes like this. . . fell down. . . whoosh. . . and zcame up” using his other hand to “draw” the z-axis rising up out of his hand that was acting as the xy-plane (SG). Tina stillvoiced her confusion (S). Next Carl drew and labeled the xy-axes on a pad of paper and stated, “Here is x–y.” then allowedthe pad to fall to his desk with a thump. Without saying anything, he then used a pen to represent the z-axis rising out ofhis drawing on the paper (ISA) and directed his gaze at Tina. . . Carl recognized Tina’s continued confusion. He then used hisgaze and a pointing gesture to indicate where the walls met and said (SGA), “in the corner of the room, z is going up . . .”

Note that when a teacher was working with a certain semiotic system and then began pointing to its elements, as wasdenoted with a GP code, the pointing was not considered a connection to a different semiotic system. Instead pointing wasseen as a means to highlight elements of the same semiotic system.

5.1. Semiotic connections featuring inscriptions

Unlike Tina in the first group, Kara in Group Two successfully made the conversion from her group’s artifact-basedrepresentation (involving a graphical model) to the inscription notation that involved ordered triples. Throughout the group’sdiscussion Kara was persistent about making sure that connections between representations were voiced.

Kara reminded the group that they were required to explain their solutions (S). Dave directed attention to the model pointingat the three axes stating that they defined the regions; he continued explaining that four of the regions existed where the
z-value was positive and four where the z-value was negative (SGPA). Kara asked (IS), “so should we say that the regionsare determined by the signs of the axes, and it would be eight separate ones if we could list them?” as she began to starther list. Both Ruth and Dave agreed and through collaborative discussion that referenced their lists came to the conclusionas each pointed to the corresponding regions on the 3D kit that two to the third power described the number of different


Table 2Teacher’s semiotic connections featuring types of semiotic resources within the K5 Diagram.

I S G A

Group OneCarl 3 0 3 1Tina 0 0 2 0Bonnie 2 2 0 0Jaime 0 0 1 0Kendra 0 2 1 0

Group TwoDave 4 0 1 2

cw

5

im

5

c

5

sk

5

I

tc

Kara 3 0 0 4Donna 5 0 0 2Ruth 2 0 2 1

“combinations” that existed for the signs of the axes (ISGP). In order to ensure that her list was complete, Kara pointed totwo of the regions on Dave’s model as she discussed the signs of each of the axes with Ruth (ISGPA).

Bonnie read the in-text question about the number of regions in three-dimensional space. Carl’s initial response was aonversion from the graphic inscription of the text to a verbalized combinatorial approach. The following excerpt relates hisritten solution.

Carl immediately responded that the answer was eight (S). When prompted by his group members to explain, he responded,“It’s the combination of two times two times two” (S). Bonnie said that she would just list the possible “combinations” (S).Carl started reciting then writing, “You can have positive-positive-positive, positive-positive-negative, positive-positive. . .”(IS) then trailed off.

.2. Semiotic connections featuring speech

The following narrative shows that in Group One Bonnie was able to immediately offer a connection from the givennscription (the figure in text) to speech using metaphor. Kendra easily extended the speech via treatment to a different

etaphor.

Jaime was selected as the initial reader for the group. She read the first section of text out loud (IS) and displayed the relatedfigures (I). Bonnie compared the negative z-axis to “digging down in the dirt” (S) and Kendra compared the positive z-axisto going up a ladder (S).

.3. Semiotic connections featuring gesture

Carl provided a connection from the given inscription to gesture using a tracing motion with his fingers. Jaime made aonnection by moving from the given inscription to an embodied gesture using three fingers.

Carl used his index fingers to trace out three orthogonal axes in the air, asking Jaime about their orientation and labeling(SG). Jaime configured three fingers of one hand (thumb, index and middle) each orthogonal to the other, using the indexfinger of the other hand to point out which finger in this gesture represented the x-, y-, and z-axes (SG).

.4. Semiotic connections featuring artifacts

Group Two’s interactions were driven by the 3D kit. Often connections occurred as Kara redirected the group to repre-entations using the 3D kit. Even though Kara made many of the connections, her speech signified both her underdevelopednowledge of 3-space and her need to mediate the meaning making process through the use of the 3D kit.

As Kara read the question involving the number of regions in 3-space, she asked Dave to use his 3D kit to model the situation(SA). Donna reread the question aloud (S) while Dave pointed to each of the four regions with positive z values on the 3D kit(GPA). Kara and Ruth copied Dave’s movements while counting aloud (SGPA). Kara said (S), “I still think it’s four [regions],aren’t they just. . . bigger?”

.5. Summary of teachers’ semiotic connections

The frequencies for the semiotic connections made by each teacher in Group One and in Group Two can be seen in Table 2.
n this section each teacher’s use of conversions is discussed.
While Carl demonstrated understanding of the content, he did not successfully communicate his ideas. Each representa-ion was presented in a staccato fashion, rapidly and without explicit connections. Carl was easily able to follow the semioticonnections of others as well as make connections himself; however, he often jumped rapidly from one semiotic system


to another without verbalizing the connections between the two. He used speech to make sound effects to accompany hisgestures, or he gave a cursory introduction to his gesture with phrases such as “It’s like a box.” However, he failed to verbalizewhy his representation was “like a box,” how it connected to the situation at hand, or how it connected to previous repre-sentations of the situation. Carl’s introduction of ordered triples and later his JELL-O® explanation (prompted by Kendra’sverbal introduction of the metaphor) are intimately related; however, he failed to capitalize on their relationship in hisexplanations to Tina.

Tina never seemed to understand the situation. Her facial expressions and her comments suggested that she was not ableto grasp any of the representations that arose from the semiotic connections made by her group members. Tina’s persistentconfusion provided evidence that she did not follow any of the semiotic connections made by her group members. Whileshe appeared to be most comfortable using inscriptions, none of the members of her group made conversions that addressedthe relationship between the ordered triples and the other representations they offered. As a result Tina was unable to listall eight ordered triples without Carl’s help.

Bonnie contributed two of the four connections to metaphor in Group One’s discussion. Bonnie verbalized that sheunderstood the connection Jamie made between the figure in the text and a three-fingered gesture even though she admittedthat she lacked the dexterity to imitate the gesture herself. Bonnie was also the first to suggest the conversion to orderedtriples.

Jaime was able to form a semiotic connection between the inscription of the figure in the text to her own three-fingeredgesture. She also was able to follow Carl’s connections as he jumped from representation to representation. At one point,she tried to explain to Tina that Carl’s representation, which made use of a pen as the z-axis and a drawing of the x- andy-axes, was “adding a new dimension.”

Kendra was able to follow Jaime’s connection from the inscribed figure in the text to the three-fingered gesture and wasable to mimic the gesture herself. Kendra was also able to make a connection from Carl’s box metaphor to her own JELL-O®

metaphor. Toward the end of the episode, Kendra made a conversion from the ordered triples to a multipart gesture in whichshe used hand movements and accompanying speech to show which regions would have positive z-values and which wouldhave negative z-values.

When prompted to do so by Kara, Dave was able to use the 3D kit to perform a connection from the inscription in thetext to a model using artifacts. Dave was particularly careful to verbalize connections between his conversions to and fromcombinatorial representations and the representations that made use of ordered triples. Dave also demonstrated flexibilityin his semiotic connections through the use of gesture as well as the use of the 3D kit and the corner of room to model thesituation.

Kara prompted the members of her group to utilize a variety of semiotic resources. Her questions were effective in elicitingthe semiotic connections that she needed in order to help her better grasp the situation. For example she suggested theconnection from the inscribed figure in the text to the use of the 3D kit. Additionally, Kara suggested the conversion from themodel on the 3D kit to a listing of ordered triples. Further, she required the members of her group to verbalize the connectionsbetween representations. Her questions created an environment that expected specific, open verbal communication amonggroup members as they made use of and moved between multiple semiotic resources.

Donna was careful to explain the connection from the figure in the text to the model on the 3D kit. Despite her attachmentto the two-dimensional drawing, she was more successful in her attempted conversion from the model on the 3D kit to thelisting of ordered triples. However, she did not demonstrate strong, fluid connections between conversions from either thedrawing or the model to the listing of ordered triples.

Ruth had no trouble making and following semiotic connections. She was the first member of Group Two to introduce aconnection from the two-dimensional figure in the text to a three-dimensional gesture that related the four regions abovethe xy-plane and the four regions below the xy-plane. Further, she made a connection from this gesture to the model on the3D kit. She verbalized the connections between the connections she made, and she was the group member who spent themost time discussing conversions to a listing of ordered triples with Kara.

5.6. Application of the K5 Diagram groups’ uses of semiotic connections

In this section the graph theory roots of the K5 Diagram are used to make evident and analyze the similarities anddifferences between the interactions of the members of Group One and the members of Group Two.

5.6.1. Group OneThe frequencies for semiotic connections for the teachers in Group One can be seen in Fig. 7. The four components of Fig. 7

show the cumulative frequency for connections that featured inscriptions, speech, gestures, and artifacts. These diagramswere used for the analyses that follow.

The interactions between the members of Group One were dominated by semiotic connections to gesture. Group One’sconnections from gestures to inscriptions and from artifacts to inscriptions (see Fig. 7, To Inscriptions) related semiotic bundles
that represented the three-dimensional axes to two-dimensional drawings that also represented the axes. The remainingtwo connections within the inscription vertex linked a combinatorial approach to a discussion of ordered triples. At no timedid the members of Group One make a direct or indirect conversion from a representation of the axes to a representation inwhich all possible ordered triples were listed.


Fig. 7. Group One’s use of semiotic connections.

vt

igtwcl

5

F

rcftwcba

ocodt3

6

grr

Fig. 8. Group Two’s use of semiotic connections.

All of Group One’s semiotic connections to speech involved metaphor. This included the connection within the speechertex from one metaphor to another. None of the group members made a direct conversion from any of these metaphorso another semiotic resource.

The majority of Group One’s semiotic connections resulted in gesture (see Fig. 7, To Gestures). These connections originatedn each of the four vertices of the K5 Diagram. Although the members of Group One made use of semiotic bundles that featuredesture often, they did not make connections from the axes, planes, and regions represented in their gestures back to eitherheir combinatorial representation or to the list of ordered triples seen in their written responses. Indeed, while thereere seven connections to gestures there were only two connections from gestures to other semiotic resources. Semiotic

onnections to artifacts occurred least often in this group. This result is unsurprising since artifacts were the semiotic resourceeast often employed by the members of Group One.

.6.2. Group TwoThe frequencies for semiotic connections for the teachers in Group Two can be seen in Fig. 8. The four components of

ig. 8 show the cumulative frequency for connections that featured inscriptions, speech, gestures, and artifacts.The interactions between the members of Group Two were dominated by connections between two types of semiotic

esources: inscriptions and artifacts. Unlike Group One, the members of Group Two made explicit direct, and indirect,onnections between representations. Three of Group Two’s connections from artifacts to inscriptions involved conversionsrom the regions of their model on the 3D kit to their listing of ordered triples. The remaining two connections form artifactso inscriptions related the model on the 3D kit to a two-dimensional drawing of the axes. Of the eight semiotic connectionsithin the inscription vertex, four related a combinatorial approach to a listing of ordered triples and the other four were

onversions that related a drawing of the axes to a listing of ordered triples. Of particular note is that although members ofoth groups discussed a combinatorial approach, only the members of Group Two related the use of 23 to the signs of thexes and the number of regions.

While the members of Group One made use of several metaphors, none of the semiotic connections made by the membersf Group Two featured speech. Rather than standing alone, gestures made by the members of Group Two mediated twoonnections from two-dimensional drawings to a representation that made use of the model on the 3D kit (semiotic chainingccurring from inscriptions to gestures to artifacts). Four of the five connections from inscriptions to artifacts related two-imensional drawings to the model on the 3D kit. The remaining connection was a conversion from inscriptions to artifactshat occurred when the members of Group Two associated each of the ordered triples with its corresponding region on theD kit.

. Discussion

The first research question considered the teachers’ use of the four observable semiotic resources found in the K5 Dia-ram as they communicated and reasoned about regions in three-dimensional space. The teachers’ discussions and writtenesponses indicated that there were multiple ways in which they communicated, conceptualized, and represented theegions in 3-space. The K5 Diagram provided unique insights into the types of semiotic resources used by the two groups


to explore 3-space. It is unsurprising that inscriptions were the most extensively used type of semiotic resource since theproblem posed and the submitted solutions to the problem were both inscriptions. What is surprising is that both groupschose to submit solutions that explained the situation using a listing of all possible ordered triples. Both groups made useof gesture; and the members of Group Two used artifacts extensively. Rather than submit solutions that mimicked thevisual nature of these representations, most teachers made use of the inscription-based ordered triple notation as their solesubmitted response.

The results suggest that many of the teachers did not initially have the subject matter knowledge one would expectfrom secondary mathematics teachers regarding the partitioning of a three-dimensional space by axes. While many wereable to collaboratively make meaning of the problem posed using a variety of semiotic resources and a recursive processthat allowed for the mutual development of understanding and communication (Roth & Pozzer-Ardenghi, 2006), othersstruggled to do so.

The second research question considered what the K5 Diagram revealed about how teachers communicated and reasonedabout regions in three-dimensional space. Semiotic resources are created and given meaning by the people using them. Asindividuals become more accustomed and adept at using a system of semiotic resources, they often feel that their “ideasabout some field of knowledge can be socially shared with the help of that system” (Moreno-Armella et al., 2008, p. 101).

The members of Group One demonstrated an understanding of the situation and made numerous attempts to discussthe problem. This group used metaphors, various multistage gestures, and the corner of the room as representations of theproblem. Yet, none of their efforts to communicate with Tina were successful since they failed to realize that the semioticresources that held meaning to them did not hold the same meaning for her. In contrast, the members of Group Two used anumber of semiotic bundles featuring a particular artifact, one that held the most meaning for Kara, and by the end of theepisode, she understood the situation.

The third research question made use of the K5 Diagram to consider how the teachers used semiotic connections to makemeaning while engaging in mathematical tasks. Some of the teachers demonstrated deeper understanding of the numberof regions in three dimensions not only through their use of semiotic resources but also through their connections, and inparticular their conversions from one mathematical form to another, between a variety of semiotic resources. Others seemedto have limited understanding of the problem and how to connect representations to make meaning of the situation, evenafter extended discussion. Kara and Tina both actively solicited help from members of their groups; however, each did so ina different manner. Kara requested that connections be verbalized during semiotic connections and that those connectionsmake use of the semiotic resources she best understood. Tina’s requests for help were less specific and failed to provide anyguidance to her group members as to the semiotic resources or the mathematical form she preferred or best understood.

Observations of Tina suggested that she was most successful when using inscriptions, namely the ordered triples; yether group members tried to respond to her needs by offering multiple representations of the situation using gestures anddrawings of the graphical situation as well as metaphors. The members of this group moved from representation to repre-sentation often failing to verbalize any connections between their semiotic bundles or between the different mathematicalforms represented. The unrealized potential of the gestures, drawings, and metaphors is marked since with some explicitspecification these semiotic resources could have helped Tina make the conversion from the spatial regions on the graphicalmodel to their respective ordered triples. Ultimately none of the group members verbalized specific connections betweenthe problem posed and the reason that ordered triples applied to the situation. This left Tina with ordered triple notationand numerous, disconnected representations of the problem without ties between them.

The members of both groups initiated and responded to each other in such a manner that suggested they were interestedin interaction and sensitive to the needs of their group members (Sfard & Kieran, 2001). Although Tina and Kara began withsimilar levels of spatial literacy, they finished with very different levels of understanding by the end of the episode. Analysisusing the K5 Diagram suggests this may have occurred because of Group One’s failure to make explicit connections betweensemiotic resources during conversions.

The fourth research question considered what the K5 Diagram revealed about the connections made between semioticresources. Those teachers who demonstrated the deepest understanding and were best able to communicate this under-standing both made connections and explicitly verbalized the connections between different mathematical forms as wellas between multiple types of semiotic resources seen in the K5 Diagram.

The K5 Diagram provided a visual representation of how the semiotic connections made by the members of Group Twofollowed one of two routes. In one route, group members began with semiotic bundles that featured an inscription (thedrawing of the axes in the text), then they made connections to semiotic bundles featuring artifacts (often first makingconnections to semiotic bundles featuring gesture) then finally making conversions to inscriptions (a listing of orderedtriples). In the second route, group members made conversions from inscriptions (either their own drawings or the one inthe text) to a different inscription (a listing of ordered triples).

In contrast to Group Two, the K5 Diagram revealed the semiotic chaining of the members of Group One often involvedconnections to speech or gesture, and rarely did these make explicit the conversions back to the ordered triples the groupmembers submitted at the end of the task. Connections were never made between the ordered triples and the unidirectional
conversions to semiotic systems and bundles that incorporated speech and gestures and featured a different mathematicalform (i.e., a graphical model of the situation). For those teachers who were most comfortable working with inscription-basedordered triples, the lack of explicit verbal connections in the conversion to a different mathematical form was detrimentalto overall understanding.

7

euomtota

itsla

R

A

AB

BB

BDD

GGHLLL

M

MM

M

NNN

P

RR

R

R

R

S

S

SST

V


. Conclusions

The K5 Diagram provides a visual representation that displays the dynamic nature of the making meaning individuals’mploy in social settings. Not only does it provide insight into two different things (the semiotic resources that individualsse and the connections groups make between them), it does so in an efficient and easy-to-read manner. The visual layoutf the vertices of the K5 Diagram affords a complete, and more importantly concise, picture of the interactions of groupembers. The visual display of information presented in the K5 Diagram would have been difficult to glean from annotated

ranscripts of the teachers’ interactions. In short, the K5 Diagram serves as an effective tool that collapses all interactions thatccurred in a linear time-continuum into a single visual representation. This simplification provided a platform that allowedhe research team to see the inflow and outflow of the semiotic resources used within the context of the entire episode by

single group. By highlighting the “flow” of a group’s discussions, it also made cross-group comparisons much easier.The intent of the researchers was to provide a method to explore, analyze, and convey meaningful information about

ndividuals, in this case teachers, interacting in a collaborative problem solving episode. The data presented here suggesthat teachers are likely to use numerous semiotic resources to communicate and reason about regions in three-dimensionalpace. The K5 Diagram served as a helpful tool guiding the semiotic analysis of the two episodes in this study, especially inight of their extended duration and the number of individuals involved. In short, the implicit flexibility of the K5 Diagramllowed analysis both of teachers’ interactions within a single group as well as between the two groups of teachers.

eferences

libali, M. W., Phillips, K. M. O., & Fischer, A. D. (2009). Learning new problem-solving strategies leads to changes in problem representation. CognitiveDevelopment, 24(2), 89–101.

rzarello, F. (2006). Semiosis as a multimodal process. Revista Latinoamericana de Investigación en Mathemática Educativa, 9, 267–299.all, D. L., & Bass, H. (2000). Interweaving content and pedagogy in teaching and learning to teach: Knowing and using mathematics. In J. Boaler (Ed.),

Multiple perspectives on the teaching and learning of mathematics (pp. 83–104). Westport, CT: Ablex.all, D. L., Thames, M. H., & Phelps, G. (2008). Content knowledge for teaching: What makes it special? Journal of Teacher Education, 59(5), 389–407.arsalou, L. W., Niedenthal, P. M., Barbey, A., & Ruppert, J. (2003). Social embodiment. In B. Ross (Ed.), The psychology of learning and motivation (pp. 43–92).

San Diego, CA: Academic.arwell, R. (2009). Researchers’ descriptions and the construction of mathematical thinking. Educational Studies in Mathematics, 72(2), 255–269.uval, R. (2006). A cognitive analysis of problems of comprehension in a learning of mathematics. Educational Studies in Mathematics, 61(1), 103–131.uval, R. (2008). Eight problems for a semiotic approach in mathematics education. In L. Radford, G. Schubring, & F. Seeger (Eds.), Semiotics in mathematics

education: Epistemology, history, classroom, and culture. Rotterdam, The Netherlands: Sense Publishers.oldin-Meadow, S., & Beilock, S. L. (2010). Action’s influence on thought: The case gesture. Perspectives on Psychological Science, 5(6), 664–674.reeno, J. (1997). On the claims that answer the wrong questions. Educational Researcher, 26(1), 5–17.oopes, J. (Ed.). (1991). Peirce on signs: Writings on semiotic by Charles Sanders Peirce.. Chapel Hill, NC: University of North Carolina Press.akoff, G., & Johnson, M. (1980). Metaphors we live by. Chicago, IL: University of Chicago Press.akoff, G., & Johnson, M. (1999). Philosophy in the flesh: The embodied mind and its challenge to western thought. New York: Basic Books.esh, R., Post, T., & Behr, M. (1987). Representations and translations among representations in mathematics learning and problem solving. In C. Janvier

(Ed.), Problems of representations in the teaching and learning of mathematics (pp. 33–40). Hillsdale, NJ: Lawrence Erlbaum Associates, Inc.cGee, D., Moore-Russo, D., Ebersole, D., & Lomen, D. (in press). Visualizing three-dimensional calculus concepts: The study of a manipulative’s effectiveness.

PRIMUS: Problems, Resources, and Issues in Mathematics Undergraduate Studies.cNeill, D. (1992). Hand and mind. Chicago: University of Chicago Press.oore-Russo, D., Conner, A., & Rugg, K. I. (2011). Can slope be negative in 3-space? Studying concept image of slope through collective definition construction.

Educational Studies in Mathematics, 76(1), 3–21.oreno-Armella, L., Hegedus, S. J., & Kaput, J. J. (2008). From static to dynamic mathematics: Historical and representational perspectives. Educational

Studies in Mathematics, 68(2), 99–111.ational Council of Teachers of Mathematics. (2000). Principles and standards for school mathematics. Reston, VA: Author.ational Council of Teachers of Mathematics. (2006). Curriculum focal points for prekindergarten through grade 8 mathematics. Reston, VA: Author.únez, R. E., Edwards, L. D., & Matos, J. F. (1999). Embodied cognition as grounding for situatedness and context in mathematics education. Educational

Studies in Mathematics, 39(1-3), 45–65.resmeg, N. (2006). Semiotics and the connections standard: Significance of semiotics for teachers of mathematics. Educational Studies in Mathematics,

61(1–2), 163–182.adford, L. (1998). On signs and representation: A cultural account. Scientia Paedagogica Experimentalis, 35(1), 277–302.adford, L. (2002). The seen, the spoken and the written: A semiotic approach to the problem of objectification of mathematical knowledge. For the Learning

of Mathematics, 22(2), 14–23.adford, L. (2008). A cultural theory of learning. In L. Radford, G. Schubring, & F. Seeger (Eds.), Semiotics in mathematics education: Epistemology, history,

classroom, and culture (pp. 215–234). Rotterdam, The Netherlands: Sense Publishers.adford, L., Bardini, C., Sabena, C., Diallo, P., & Simbagoye, A. (2005). On embodiment, artifacts, and signs: A semiotic-cultural perspective on mathematical

thinking. In H. L. Chick, & J. L. Vincent (Eds.), Proceedings of the 29th conference of the international group for the psychology of mathematics education (pp.113–120). Melbourne: PME.

oth, W.-M., & Pozzer-Ardenghi, L. (2006). Tracking situated, distributed, and embodied communication in real time. In M. A. Vanchevsky (Ed.), Focus oncognitive psychology research (pp. 237–261). Hauppauge: Nova Science.

choenfeld, A. H., Smith, J. P., & Arcavi, A. (1993). Learning: The microgenetic analysis of one student’s evolving understanding of a complex subject matterdomain. In R. Glaser (Ed.), Advances in instructional psychology (pp. 55–175). Hillsdale, NJ: Erlbaum.

fard, A., & Kieran, C. (2001). Cognition as communication: Rethinking learning-by-talking through multi-faceted analysis of students’ mathematicalinteractions. Mind, Culture, and Activity, 8(1), 42–76.

hulman, L. S. (1986). Those who understand: Knowledge growth in teaching. Educational Researcher, 15(2), 4–14.mith, L. B. (2005). Cognition as a dynamic system: Principles from embodiment. Developmental Review, 25, 278–298.hom, J. S., & Roth, W.-M. (2011). Radical embodiment and semiotics: Toward a theory of mathematics in the flesh. Educational Studies in Mathematics,

77(2-3), 267–284.an Leeuwen, T. (2005). Introducing social semiotics. London: Routledge.

Documents

Using the K5 Connected Cognition Diagram to analyze teachers’ communication and understanding of regions in three-dimensional space