Volume 6. Constructing Dynamic Triangles Together: The Development of Mathematical Group Cognition
This book analyzes the work of a team of three students as they become introduced to dynamic geometry during eight hour-long online sessions using VMT with GeoGebra.
Published in November 2015 by Cambridge University Press in their book series, "Learning in doing: Social, cognitive and computational perspectives."
Stahl, G. (2016). Constructing dynamic triangles together: The development of mathematical group cognition. Cambridge, UK: Cambridge University Press. Learning in doing: Social, cognitive and computational perspectives book series. 250 pages.
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This book builds on earlier publications about the Virtual Math Teams (VMT) Project, realizing their arguments, documenting their claims, fleshing out their theory and fulfilling their promises. It is the culminating volume of a cycle of books reporting on the Project:
• Group Cognition (Stahl, 2006, MIT Press) introduced the VMT Project as a response to issues in CSCL and CSCW. It outlined a theory of group cognition as a framework appropriate to computer-mediated interaction. It recommended adapting methods of Conversation Analysis to online text chat. It proposed that the analysis of group processes could provide insight into online collaborative learning.
Providing computer support for the learning of mathematics is a major educational challenge today. Networked computers provide an opportunity to explore collaborative-learning approaches. The recent availability of dynamic-geometry software provides further opportunity for innovative approaches to math education. This book reports on an extensive research effort involving teaching teachers and their students about dynamic geometry in an online collaboration environment. Specifically, it documents the cognitive development of a team of three students learning in that online social setting. The extended case study shows how the team enacted the tools and adopted group practices within an educational research project, which was designed to extend and support their ability to collaborate, to engage in mathematical discourse and to explore or construct dynamic-geometric figures. As a whole, the book provides detailed empirical support for the theory and practice of group cognition.
• Studying Virtual Math Teams (Stahl, 2009, Springer) described the technology approach and affordances of the VMT software environment. It extended the author’s theory of group cognition. It provided illustrative analyses of brief excerpts of student interaction in VMT.
•Translating Euclid (Stahl, 2013, Morgan & Claypool) reviewed the VMT Project, its technology, curriculum, design-based approach and educational goals. It discussed the integration of multi-user dynamic geometry into the VMT environment. It further elaborated the theory of group cognition.
The book documents the findings of the Virtual Math Teams (VMT) Project, a paradigmatic example of Computer-Supported Collaborative Learning (CSCL) exploration, incorporating a unique model of Human-Computer Interaction (HCI) analysis. Directed by the author for the past 12 years, the VMT Project pioneered a method of analyzing interaction data, adapting ethnomethodologically inspired Conversation Analysis to the special conditions of computer-mediated collaboration and to the needs of design-based research in math education. This fine-grained report on the VMT Project applies its methods longitudinally to the full eight hours of one student group’s interaction. In this analysis effort, it details the team’s cognitive development. It ties the development of their group cognition to the technological mediation, which takes place at multiple levels of the project:
• The students interact exclusively through the VMT online collaboration environment using text chat.
The centrality of computer support to the project makes this book relevant to (i) CSCL, (ii) HCI and (iii) educational technology:
• The students explore dynamic geometry in a computer simulation.
• The domain of dynamic geometry only exists in computer media.
• The curriculum is technologically scripted without any teacher intervention.
• All the data is collected electronically through comprehensive instrumentation.
(i) From a CSCL perspective, the book is unique in providing an analysis of small-group cognitive development in terms of the adoption of group practices. It not only documents that learning took place by the student team, but also details how the learning took place by observing the enactment of numerous group practices. It provides a rich picture of learning on many levels, not just a single learning outcome. The cognitive development of the observed team of students is conceived as computer-supported collaborative learning, in which learning is primarily viewed at the group unit of analysis of collaboration and all the communication takes place through computer-mediated interaction. This approach is framed in the philosophy of group cognition, which has emerged from the VMT Project and is grounded in its findings.
(ii) From an HCI standpoint, the book’s analysis is paradigmatic in that it documents an investigation in which interaction analysis played a central role in the design-based research process, providing feedback to the project at multiple points (advice to teachers between sessions, revisions for the next cycle and formative evaluation of the overall project, including elaboration of the theoretical framework).
(iii) From an educational technology approach, the book is distinctive in offering a longitudinal case study, which details the cognitive development starting when the students first encounter online collaborative dynamic geometry. It identifies dozens of group practices by which the team of students learns to collaborate, to enact software tools, to understand geometric figures and to discuss mathematical invariants and dependencies. It thereby shows how an online collaboration environment can facilitate learning, specifically the critical development of geometric reasoning, by providing a supportive space for the adoption of group practices.
As the concluding volume reporting on the VMT Project, this book illustrates a successful implementation of group-cognition research and analysis. Since it was proposed in the 2006 volume, the theory of group cognition has been increasingly accepted as an alternative to the traditional educational-psychology approach to instructional technology, focused on measurable learning outcomes of individual minds. As a presentation of CSCL methodology, the book provides an alternative or potential complement to statistical coding approaches. Within HCI, it shows that an ethnomethodologically informed approach can generate implications for design systematically within a practical design-based research process.
Constructing Triangles Together: The Development of Mathematical Group Cognition completes the story of the VMT Project. It centers on an extended case study: the detailed longitudinal analysis of eight hours of interaction by a virtual math team of three middle-school girls working on an introductory sequence of dynamic-geometry challenges. It fulfills the promises and claims of previous publications on VMT by demonstrating the success of the methods they proposed, and carrying out systematic analysis of a team’s entire online collaborative-learning experience. Along the way, it provides lessons for online curricular design, for CSCL technology and for HCI analysis. It also fills in the theory of group cognition with concrete results based on detailed data showing how collaborative learning takes place through the enactment of specific group practices for collaboration, math discourse and software tool usage.
table of contents
The bulk of the volume conducts a fine-grained analysis of the student interaction and identifies the team’s adoption of group practices in their interaction. The analysis chapters illustrate many aspects of sequential interaction analysis, show how the students enact the use of the available technology, examine the student interpretation of curricular artifacts and display the student engagement in specific group practices. Each analysis chapter concludes with an assessment of the team’s cognitive development and a set of implications for redesign of project details. This core of the book is preceded by methodological considerations and followed by theoretical reflections:
• Researching Mathematical Cognition. The opening chapter emphasizes the importance of mathematical cognition in the modern world and the difficulty it presents for many students. It briefly considers issues of schooling and theories concerning the development of mathematical understanding. It then proposes a case-study approach, incorporating sequential interaction analysis. Building on Vygotsky’s ideas, it suggests focusing on developmental processes at the group unit of analysis.
• Analyzing the Development of Group Cognition. The VMT Project is described as design-based research, which incorporates cycles of refining technology, curriculum and theory through iterative trials with classroom teachers and students. The goals of the project and of the analysis in this book include: development of collaboration skills, mathematical discourse and usage of software tools. Dynamic geometry is briefly described, with its characteristics of dragging, constructing and defining mathematical dependencies. The analytic methodology is then presented as sequential interaction analysis, with a special emphasis on long sequences of interaction involved in geometric problem solving. Such analysis can highlight the display by students of their collaborative mathematical development as they chat, manipulate graphical objects, explore problems, construct geometric figures and articulate solutions. In this way, analysis not only indicates that certain learning took place, but also shows how it took place. The development of mathematical cognition by the group can thereby be documented in terms of group practices that the team adopts.
• Topic 1: The Team Develops Collaboration Practices. This first analysis chapter shows how the group of three students developed into a collaborative team, largely during their initial hour together online. At first, the students had no idea what to do in the VMT environment. However, they successively responded to suggestions within the environment—textual instructions, software displays, results of explorations. The chapter enumerates many specific group collaboration practices that they adopted in their first session, which served them well for the rest of their work together.
• Topic 2: The Team Develops Dragging Practices. Dragging points of geometric figures and observing the consequent changes is a central characteristic of dynamic geometry. Dragging can be used for a variety of purposes, such as aligning parts of a geometric figure, exploring a construction or testing if dependencies hold during dragging. In their second session, the team developed a variety of group practices related to dynamic-geometry dragging.
• Topic 3: The Team Develops Construction Practices. Construction is a conventional focus in learning Euclidean geometry. In this session, the team engages in several traditional construction tasks. In the process, they adopt a number of group construction practices that are special to dynamic geometry. The chapter also investigates problems the team had in constructing figures, how they overcame some of their problems and how they missed opportunities that had been designed into the tasks. During this session, the team displayed significant progress in moving from a visual to a more formal mathematical conceptualization of construction.
• Topic 4: The Team Develops Tool-Usage Practices. In this session, the team hones its skills using the dynamic-geometry tools, including the procedure to define new custom tools. The team adopts some group practices for using the tools.
• Topic 5: The Team Identifies Dependencies. This chapter explores in even greater detail a particularly exciting developmental breakthrough by the team. Viewed superficially, the team seems to be floundering with a challenging problem involving inscribed triangles. They seem to have digressed even in their collaboration practices. However, in the end of the session it appears that the student who often seems to be the weakest in mathematical understanding solves the problem. The particular geometry task is one that has been used often in the VMT Project and is rarely solved within an hour, even by mathematically experienced adults. A close analysis in this chapter shows how the team actively explored the problem and potential solution techniques through extensive investigation of dragging and construction approaches. The eventual solution actually involved contributions from all three team members and displayed a clear understanding of the solution logic.
• Topic 5 Continued: The Team Constructs Dependencies. The team was given another hour-long session to tackle a related dynamic-geometry problem. This time, the triangles were replaced by squares. The team had not worked with constructing squares before, but eventually arrived at an elegant solution for doing that. Once they constructed the outside square, the whole team immediately expressed knowledge of how to construct an inscribed square in it. This displayed their firm understanding of what they had accomplished in the previous session with the triangles. Their success also confirmed their impressive development of mastery of collaboration, dragging, construction, tool-usage and dependency practices.
• Topic 8: The Team Uses Transformation Tools. For their next session, the teacher skipped ahead to an introduction to unrelated tools for rigid transformations (translation, reflection, rotation). Although the team had some success here, they failed to gain much insight into the transformation paradigm of constructing dependencies. Here, the analysis revealed the need for considerably more curricular support.
• Topic 13: The Team Develops Mathematical Discourse Practices. The team’s final session involved the exploration of many different quadrilaterals, to determine dependencies in their construction through dragging. The team explored seven figures, with very different results. Some figures were too simple and others too difficult to understand through a couple minutes of dragging. However, in working on Poly2, the team engaged in impressive dragging and in striking mathematical discourse about dependencies. This session displayed both the extent of the team’s development along multiple dimensions and the fragility of this development. The analysis of the team’s interaction suggests revisions to the curriculum for future research trials.
• Contributions to a Theory of Mathematical Group Cognition. In this theoretical chapter, the findings of the preceding analysis are reflected upon as aspects of the theory of group cognition, specifically as applied to mathematics. The sequences of group practices adopted by the team of students are conceptualized in the light of contemporary cognitive theory. For instance, the group collaborative practices are seen as contributing to a sense of group agency, using insights from Latour and others. The mathematical discourse practices are confronted with conclusions of Sfard. Group tool-usage practices are considered in terms of Rabardel’s concept of instrumental genesis. Dragging is related to embodied group cognition; construction to situated group cognition; and dependencies to designing.
•Constructing Triangles Together. The concluding chapter has three parts. First, it considers the development of mathematical cognition as a dialectical process. Then it recaps the book’s implications for re-design of the VMT collaboration environment, especially the curriculum of dynamic-geometry tasks. Finally, it reviews what has been learned from the VMT Project and suggests prospects for future efforts along these lines.