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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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* Download PDF free for reading online or printing: analysis.pdf

Note: The pre-publication materials were last revised March 16, 2015, from the final manuscript. This is a pre-publication version of the book. This version has not been edited, laid out or paginated by Cambridge University Press. Please do not cite page numbers from this version or quote from it. This version is only for informal use and may not be duplicated. Please refer to the published version for official usage, citation and pagination.


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: 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.

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 centrality of computer support to the project makes this book relevant to (i) CSCL, (ii) HCI and (iii) educational technology:

(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: