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Volume 5. Translating Euclid: Creating a Human-Centered Mathematics

Available now in e-book & in paperback!

Stahl, G. (2013). "Translating Euclid: Creating a Human-Centered Mathematics." Morgan & Claypool Publishers. 221 pages.

* Order from Morgan Claypool Publishers: http://dx.doi.org/10.2200/S00492ED1V01Y201303HCI017

* Order from Amazon: http://www.amazon.com/s/ref=ntt_at_ep_srch?ie=UTF8&search-alias=books&field-author=Gerry+Stahl

Note:You and your students can download a free e-copy from the Morgan Claypool site if your university subscribes to the Synthesis Lectures on Human-Centered Informatics series. Hundreds of university libraries around the world subscribe (for a list see http://www.morganclaypool.com/page/licensed)

Students, faculty and staff at Drexel can use this link for a free copy of the e-book: http://records.library.drexel.edu/record=b2145294~S9 When you click for resource, login with your DrexelOne id.

For more information on the book, see: http://www.morganclaypool.com/doi/abs/10.2200/S00492ED1V01Y201303HCI017

Download a free copy of Chapter 1, which contains an overview of the whole book: http://www.morganclaypool.com/doi/suppl/10.2200/S00492ED1V01Y201303HCI017


Translating Euclid reports on an effort to transform geometry for students from a stylus-and-clay-tablet corpus of historical theorems to a stimulating computer-supported collaborative-learning inquiry experience.
The origin of geometry was a turning point in the pre-history of informatics, literacy and rational thought. Yet, this triumph of human intellect became ossified through historic layers of systematization, beginning with Euclid’s organization of the Elements of geometry. Often taught by memorization of procedures, theorems and proofs, geometry in schooling rarely conveys its underlying intellectual excitement. The recent development of dynamic-geometry software offers an opportunity to translate the study of geometry into a contemporary vernacular. However, this involves transformations along multiple dimensions of the conceptual and practical context of learning.
Translating Euclid steps through the multiple challenges involved in redesigning geometry education to take advantage of computer support. Networked computers portend an interactive approach to exploring dynamic geometry as well as broadened prospects for collaboration. The proposed conception of geometry emphasizes the central role of the construction of dependencies as a design activity, integrating human creation and mathematical discovery to form a human-centered approach to mathematics.
This book chronicles an iterative effort to adapt technology, theory, pedagogy and practice to support this vision of collaborative dynamic geometry and to evolve the approach through on-going cycles of trial with students and refinement of resources. It thereby provides a case study of a design-based research effort in computer-supported collaborative learning from a human-centered informatics perspective.


Computer-Supported Collaborative Learning (CSCL), Design-Based Research (DBR), Virtual Math Teams (VMT), group cognition, dynamic geometry, post-cognitive philosophy, interaction analysis, creative discovery, multi-user software, interactional resources.

download the pre-publication version here

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

Note: The following materials were last revised March 16, 2013, from the final manuscript. This is a pre-publication version of the book. This version has not been edited, laid out or paginated by Morgan & Claypool. 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.


Chapter 1. Vision: The Cognitive Potential of Collaborative Dynamic Geometry
Chapter 2. History: The Origin of Geometry
Chapter 3. Philosophy: The Obfuscation of Geometry
Chapter 4. Mathematics: Demythologizing Geometry
Chapter 5. Technology: Deconstructing Geometry
Chapter 6. Collaboration: Group Geometry
Chapter 7. Research: Analyzing Geometry
Chapter 8. Theory: Resources for Geometry
Chapter 9. Pedagogy: Designing Geometry
Chapter 10. Practice: Doing Geometry
Chapter 11. Design-Based Research: Human-Centered Geometry
Author Index
Author’s Biography


The following review was for a chapter published in the International Handbook of Collaborative Learning that was very similar in content to parts of Chapter 3 of Translating Euclid.

Book Review: Learning With and From Others: A Review of The International Handbook of Collaborative Learning

The International Handbook of Collaborative Learning. Cindy E. Hmelo-Silver, Clark A. Chinn, Carol K. K. Chan, & Angela O’Donnell (Eds.) (2013). New York, NY: Routledge, 516 pp. ISBN: 978-0-415-80573-5 (hb) $295. ISBN: 978-0-415-80574-2 (pb) $114.95.

Reviewed by Alan H. Schoenfeld, University of California at Berkeley

Chapter 4, “Theories of Cognition in Collaborative Learning,” by Gerry Stahl, provides an acute commentary on the state of the art. Stahl focuses on computer-supported collaborative learning (CSCL), but his findings are more general.

Stahl begins with a terse but useful historical and philosophical summary, tracing the evolution of theoretical perspectives that focus on the individual mind: (from Socrates through Kant and Husserl) to theories that are social in character (from Hegel’s conceptualization of the individual as a social being to the critical social theory, existential phenomenology and linguistic analysis engendered respectively by Marx, Heidegger, and Wittgenstein). These, in turn, provide the underpinnings for the approaches of contemporary theorists such as Latour, Engeström, Lave, Hutchins, Suchman, Kling and Schegloff.

What I find most useful about Stahl’s chapter, however, is a critique that resonates deeply with my own sense of what matters—a need for theory-based explanations at a level of mechanism with careful specification of the unit(s) of analysis. In critiquing the current state, Stahl also points the way forward:

The CSCL-related literature on small groups and on post-cognitive phenomena provide some nice studies of the pivotal role of small groups, but they rarely account for this level of description theoretically. They are almost always in the final analysis based on either a psychological view of mental processes at the individual level or a sociological view of rules at the community level. They lack a foundational conception of small groups as a distinct level of analysis and description. They often confuse analysis at the small-group level and at the societal level, and they lack a developed account of the relationships among the individual, small group, and community of practice. Yet, there are distinct phenomena and processes at each of these levels, and analyses at different levels of description reveal different insights. (pp. 83–84)

Although Stahl restricts his comments to CSCL, his comments apply much more broadly. Reflecting (and acting) on these issues would be good for all empirical researchers.

"videos related to "Translating Euclid"


YouTube link: http://youtu.be/h66yKE48atE (50 min)

Lecture given by Gerry Stahl at the LINCS Center at the University of Gothenburg, Sweden, October 9, 2013. Presentation on the process of designing the technology, curriculum and research reported in "Translating Euclid".

Abstract: How should one translate the classic-education approach of Euclid’s geometry into the contemporary vernacular of social networking, computer visualization and discourse-centered pedagogy? How should one design the technology, mathematics, resources and presentation to incorporate collaborative discourse, computer manipulation and hands-on experience to support deep learning of mathematical thinking? What design-based-research approach can be used to explore and assess collaborative learning of dynamic geometry? (See Ch. 1, 5, 6, 11.)

Download of Powerpoint slides for presentation: http://GerryStahl.net/pub/designing.pdf

Alternative downloads of video: http://GerryStahl.net/elibrary/euclid/designing.mp4 (400 mB), http://GerryStahl.net/elibrary/euclid/designing_hd.mp4 (2,350 mB), http://GerryStahl.net/elibrary/euclid/designing_fullhd.mp4 (12,200 mB)

Discussion of Designing

YouTube link: http://youtu.be/Os75d3LFhk4 (52 min)

Discussion session following lecture given by Gerry Stahl at the LINCS Center at the University of Gothenburg, Sweden, October 9, 2013, on the process of designing the technology, curriculum and research reported in "Translating Euclid".

Alternative download of video: http://GerryStahl.net/elibrary/euclid/designing_discussion.mp4 (400 mB)


YouTube link: http://youtu.be/GDVO4Q4B6ss (1 hour)

Lecture given by Gerry Stahl at the LINCS Center at the University of Gothenburg, Sweden, October 10, 2013. Presentation on the curriculum resources reported in "Translating Euclid".

Abstract: Euclidean geometry has trained students for over 2,000 years in rational, deductive thinking and mathematical practices. Computer-based dynamic geometry (Geometer’s Sketchpad, Cabri, GeoGebra, etc.) adds three dimensions: dynamic dragging, dynamic construction (including programming custom tools) and dynamic dependencies. With these, students can drag figures to discover dependencies as well as construct figures to create dependencies. Rather than accepting geometric propositions as otherworldly truths, students can now conceive them as results of their own “creative discovery” within a local knowledge community. How can geometry education be structured to emphasize this? (See Ch. 4, 5, 9, 10.)

Download of Powerpoint slides for presentation: http://GerryStahl.net/pub/didactics.pdf

Alternative downloads of video: http://GerryStahl.net/elibrary/euclid/didactics.mp4 (574 mB), http://GerryStahl.net/elibrary/euclid/didactics_hd.mp4 (3,000 mB),

Discussion of Didactics

YouTube link: http://youtu.be/Uvgb1Fjsx_Y (16 min)

Discussion session following lecture given by Gerry Stahl at the LINCS Center at the University of Gothenburg, Sweden, October 10, 2013, on the curricular resources reported in "Translating Euclid".

Alternative download of video: http://GerryStahl.net/elibrary/euclid/didactics_discussion.mp4 (142 mB)


YouTube link: http://youtu.be/6o1AlZlVbL4 (3 hours)

Data session led by Gerry Stahl with the NAIL group at the University of Gothenburg, Sweden, October 11, 2013. Analysis of the interaction data reported in "Translating Euclid".

Abstract: Students learning collaborative dynamic geometry must acquire various practical, interactive and cognitive skills. The data sessions will focus on logs taken from a sequence of eight hour-long online chat sessions involving three middle-school students recorded in Spring 2013. The students interacted in the Virtual Math Teams (VMT) collaboration environment incorporating the GeoGebra dynamic-mathematics software. Each session was guided by a written topic involving GeoGebra tasks. The logs capture all chat posts and GeoGebra actions (opening a GeoGebra tab, selecting a GeoGebra tool, creating or dragging a GeoGebra object, etc.). The logs will be available as spreadsheets that can be configured and filtered in useful ways. In addition, the sessions can be replayed in a digital replayer system, allowing for detailed study of actions and interactions. For the workshop, a series of excerpts will be selected from the eight sessions, chosen to capture changes in the student team’s ability to engage in collaboration, software usage and geometry task accomplishment. Analysis will aim to document how the team learned the underlying practices of engaging in dynamic geometry. (See Ch. 7.)

Download of Powerpoint slides for presentation: http://GerryStahl.net/pub/analysis.pdf

Alternative download of video: http://GerryStahl.net/elibrary/euclid/analysis.mp4 (1,410 mB).

Invited lecture at Rutgers 2012

YouTube link: http://youtu.be/vogBZjdniME (lecture version, 78 min)

YouTube link: http://youtu.be/GkwI7JBq1ZQ (alternate version, 106 min)

"Translating Euclid into CSCL: Issues in design-based research on computer-supported collaborative learning of significant mathematical discourse"

The Distinguished Lecture by Gerry Stahl in the School of Communication and Information, Rutgers University, New Brunswick, NJ. February 24, 2012.

Download of Powerpoint slides for presentation: http://GerryStahl.net/pub/rutgers2012.ppt.pdf.