The Integration of Mathematics Discourse, Graphical Reasoning and Symbolic Expression by a Virtual Math Team

Learning mathematics involves mastering specific forms of social practice. In this chapter, we describe socially situated, interactional processes involved with collaborative learning of mathematics online. We provide a group-cognitive account of mathematical understanding in an empirical case study of an online collaborative learning environment called Virtual Math Teams. The chapter looks closely at how an online small group of mathematics students coordinates their collaborative problem solving using chat, shared drawings and mathematics symbols. Our analysis highlights the methodic ways group members enact the affordances of their situation (a) to display their reasoning to each other by co-constructing shared mathematical artifacts and (b) to coordinate their actions across multiple interaction spaces to relate their narrative, graphical and symbolic contributions while they are working on open-ended mathematics problems. In particular, we identify key roles of referential and representational practices in the co-construction of deep mathematical understanding at the group level, which is achieved through methodic uses of the environment’s features to coordinate narrative, graphical and symbolic resources. Mathematical Practices Developing pedagogies and instructional tools to support learning mathematics with understanding is a major goal in Mathematics Education (CCSSI, 2011; NCTM, 2000). A common theme among various characterizations of mathematical understanding in the mathematics education literature involves constructing relationships among mathematical facts and procedures (Hiebert & Wearne, 1996). In particular, recognition of connections among multiple realizations of a mathematics concept encapsulated in various inscriptional forms is considered as evidence of deep understanding of that subject matter (Kaput, 1998; Sfard, 2008; Healy & Hoyles, 1999). For instance, the concept of function in the modern mathematics curriculum is introduced through its graphical, narrative, tabular and symbolic realizations. Hence, a deep understanding of the function concept is ascribed to a learner to the extent he/she can demonstrate how seemingly different graphical, narrative and symbolic forms are interrelated as realizations of each other in specific problem-solving circumstances that require the use of functions. On the other hand, students who demonstrate difficulties in realizing such connections are considered to perceive actions associated with distinct forms as isolated sets of skills, and hence are said to have a shallow understanding of the subject matter (Carpenter & Lehrer, 1999). Reflecting on one’s own actions and communicating/articulating mathematical rationale are considered as important activities through which students realize connections among seemingly isolated facts and procedures in mathematics education theory (Sfard, 2002; Hiebert et al., 1996). Such activities are claimed to help learners notice broader structural links among underlying concepts, reorganize their thoughts around these structures, and hence develop their understanding of mathematics (Carpenter & Lehrer, 1999; Skemp, 1976). Consequently, collaborative learning in peer-group settings is receiving increasing interest in mathematics education practice due to its potential for promoting student participation and creating a natural setting where students can explain their reasoning and benefit from each others’ perspectives (Barron, 2003). Representational capabilities offered by Information and Communication Technologies (ICT) provide important affordances for exploring connections among different realizations of mathematical objects. Dynamic geometry applications like Cabri, Geometer’s Sketchpad, GeoGebra (Goldenberg & Cuoco, 1998); algebra applications such as Casyospee (Lagrange, 2005), or statistical modeling and exploratory data analysis tools like TinkerPlots (Konold, 2007) provide representational capabilities and virtual manipulatives that surpass what can be done with conventional methods of producing mathematical inscriptions in the classroom (Olive, 1998). In addition to this, widespread popularity of social networking and instant messaging technologies among the so-called Net Generation requires designers of educational technology to think about innovative ways for engaging the new generation of students with mathematical activity (Lenhart et al., 2007). Therefore, bringing the representational capabilities of existing mathematical packages together with communicational affordances of socialnetworking/messenger software can potentially support the kinds of interactions that foster deeper understanding of mathematics. Computer-Supported Collaborative Learning (CSCL) is a research paradigm in the field of Instructional Technology that investigates how such opportunities can be realized through carefully designed learning environments that support collective meaning-making practices in computer-mediated settings (Stahl, Koschmann, & Suthers, 2006). Multimodal interaction spaces—which typically bring together two or more synchronous online communication technologies such as text-chat and a shared graphical workspace—have been widely employed in CSCL research and in commercial collaboration suites such as Elluminate and BlackboardWimba to support collaborative-learning activities of small groups online (Dillenbourg & Traum, 2006; Suthers et al., 2001). The way such systems are designed as a juxtaposition of several technologically independent online communication tools not only brings various affordances (i.e., possibilities for and/or constraints on actions), but also carries important interactional consequences for the users (Cakir, Zemel & Stahl, 2009; Suthers, 2006; Dohn 2009). Providing access to a rich set of modalities for action allows users to demonstrate their reasoning in multiple semiotic forms. However, the achievement of connections that foster the kind of mathematical understanding desired by mathematics educators is conditioned upon team members’ success in devising shared methods for coordinated use of these resources (Mühlpfordt & Stahl, 2007). Although CSCL environments with multimodal interaction spaces offer rich possibilities for the creation, manipulation, and sharing of mathematical artifacts online, the interactional organization of mathematical meaning-making activities in such online environments is a relatively unexplored area in CSCL and in mathematics education. In an effort to address this gap, we have designed an online environment with multiple interaction spaces called Virtual Math Teams (VMT), which allows users to exchange textual postings as well as share graphical contributions online (Stahl, 2009). The VMT environment also provides additional resources, such as explicit referencing and special awareness markers, to help users coordinate their actions across multiple spaces. Of special interest to researchers, this environment includes a Replayer tool to replay a chat session as it unfolded in real time and inspect how students organize their joint activity to achieve the kinds of connections indicative of deep understanding of mathematics (Stahl, 2011). In this chapter we focus on the interactional practices through which VMT participants achieve the kinds of connections across multiple semiotic modalities that are indicative of deep mathematical understanding. In particular, the chapter will look closely at how an online small group of mathematics students coordinated their collaborative problem solving using digital text, drawings and symbols. We take the mathematics-education practitioners’ account of what constitutes deep learning of mathematics as a starting point, but instead of treating understanding as a mental state of the individual learner that is typically inferred by outcome measures, we argue that deep mathematical understanding can be located in the practices of collective multimodal reasoning displayed by groups of students through the sequential and spatial organization of their actions (Stahl, 2006). In an effort to study the practices of multimodal reasoning online, we employ an ethnomethodological case-study approach and investigate the methods through which small groups of students achieve joint attention to particular mathematical features of their representations in order to ground their co-construction of shared mathematical meaning (Sarmiento & Stahl, 2008, Stahl, et al., 2011). Our analysis of the excerpts presented below has identified key roles of referential and representational practices in the co-construction of deep mathematical understanding at the group level, which is elaborated further in the discussion section. Data Collection & Methodology The excerpts analyzed in this chapter are obtained from a problem-solving session of a team of three upper-middle-school students who participated in the VMT Spring Fest 2006. This event brought together several teams from the US, Singapore and Scotland to collaborate on an open-ended mathematics task on combinatorial patterns. Students were recruited anonymously through their teachers. Members of the teams generally did not know each other before the first session. Neither they nor we knew anything about each other (e.g., age or gender) except chat screen names and information that may have been communicated during the sessions. Each group participated in four sessions during a two-week period, and each session lasted over an hour. Each session was moderated by a Math Forum member; the facilitators’ task was to help the teams when they experienced technical difficulties, not to participate in the problem-solving work. During their first session, all the teams were asked to work on a particular pattern of squares made up of sticks (see Figure 1). For the remaining three sessions the teams were asked to come up with their own stick patterns, describe the patterns they observed as mathematical formulae, and share their observations with other teams through a wiki page. (1) 4 sticks, 1 square (2) 10 sticks, 3 squares (3) 18 sticks, 6 squares N Sticks Squares