Abstract
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.
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Notes
- 1.
The referential links used by the students to connect their messages to previous messages are displayed in the right-most column in the excerpts. For instance, line 745 includes Message #742 in the right-most column. This indicates that message 745 was linked to 742 by its contributor (i.e. Nan in this case). References to whiteboard objects are also marked in this column (e.g. see Fig. 6). Whiteboard drawing actions are described in bold-italics to separate them from chat messages. Note that chat postings and whiteboard drawings often interleave each other.
- 2.
Phrases quoted from chat messages are printed in bold to highlight the terms used by the participants.
- 3.
There is a parallel conversation unfolding in chat at this moment between the facilitator (Nan) and Jason about an administrative matter. Lines 740, 743, 744, and 745 are omitted from the analysis to keep the focus on the math problem solving.
- 4.
137’s referential work involves multiple objects in this instance. Although the referencing tool of VMT can be used to highlight more than one area on the whiteboard, this possibility was not mentioned during the tutorial and hence was not available to the users. Although the explicit referencing tool of the system seemed to be inadequate to fulfill this complicated referential move, 137 achieves a similar referential display by temporally coordinating his moves across both interaction spaces and by using the plural deictic term “those” to index his recent moves.
- 5.
We have observed that students use “those” (or “that”) in chat to reference items already existing in the whiteboard, but “these” (or “this”) to reference items that they are about to add to the whiteboard.
- 6.
The token “wait” is used frequently in math problem-solving chats to suspend ongoing activity of the group and solicit attention to something problematic for the participant who uttered it. This token may be used as a preface to request explanation (e.g., wait a minute, I am not following, catch me up) or to critique a result or an approach as exemplified in this excerpt.
- 7.
Goodwin (1996) proposes the term prospective indexicals for those terms whose sense is not yet available to the participants when it is uttered, but will be discovered subsequently as the interaction unfolds. Recipients need to attend to the subsequent events to see what constitutes a “pattern” in this circumstance.
- 8.
Hanks proposes the notion of indexical symmetry to characterize the degree to which the interactants share, or fail to share, a common framework relative to some field of interaction on which reference can be made. In particular, “…the more interactants share, the more congruent, reciprocal and transposable their perspectives, the more symmetric is the interactive field. The greater the differences that divide them, the more asymmetric the field.” (Hanks, 2000, p. 8.). These excerpts show that mathematical terms are inherently indexical. Establishing a shared understanding of such indexical terms require collaborators to establish a reciprocity of perspectives towards the reasoning practices displayed/embodied in the organization of the texts and inscriptions in the shared scene (Zemel & Çak\( {\i} \)r, 2009).
- 9.
- 10.
See footnote to line 746 on the use of “these” and “those”. The consistency of the usage of these terms for forward and backward references from the narrative chat to the graphical whiteboard suggests an established syntax of the relationships bridging those interaction spaces within the temporal structure of the multi-modal discourse.
- 11.
The session was scheduled to end at 7 p.m., yet the students were allowed to continue if they wished to do so. In this case Jason informed the facilitators in advance that he had to leave at 7 p.m. Central (the log is displayed in US Eastern time).
- 12.
The facilitator opens the possibility to end the session in line 855. The facilitator takes the sustained orientation of the remaining team members to the problem as an affirmative answer and lets the team continue their work.
- 13.
Sfard (2008) describes saming as the process of “…assigning one signifier (giving one name) to a number of things previously not considered as being the same” (p. 302).
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Çakır, M.P., Stahl, G. (2013). The Integration of Mathematics Discourse, Graphical Reasoning and Symbolic Expression by a Virtual Math Team. In: Martinovic, D., Freiman, V., Karadag, Z. (eds) Visual Mathematics and Cyberlearning. Mathematics Education in the Digital Era, vol 1. Springer, Dordrecht. https://doi.org/10.1007/978-94-007-2321-4_3
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