# American Institute of Mathematical Sciences

July  2010, 14(1): 17-40. doi: 10.3934/dcdsb.2010.14.17

## Community resilience in collaborative learning

 1 California State University, Channel Islands, One University Drive, Camarillo CA 93012, United States 2 Departments of Mathematics and Curriculum & Instruction, University of Texas at Arlington, Box 19408, Arlington, TX 76019-0408, United States 3 Mathematical, Computational Modeling Sciences Center, PO Box 871904, Arizona State University, Tempe, AZ 85287 4 Mathematical & Natural Sciences Division, Arizona State University, Mail Code 2352, P.O. Box 37100, Phoenix, AZ 85069-7100, United States

Received  July 2009 Revised  December 2009 Published  April 2010

This paper introduces a simplified dynamical systems framework for the study of the mechanisms behind the growth of cooperative learning in large communities. We begin from the simplifying assumption that individual-based learning focuses on increasing the individual's "fitness" while collaborative learning may result in the increase of the group's fitness. It is not the objective of this paper to decide which form of learning is more effective but rather to identify what types of social communities of learners can be constructed via collaborative learning. The potential value of our simplified framework is inspired by the tension observed between the theories of intellectual development (individual to collective or vice versa) identified with the views of Piaget and Vygotsky. Here they are mediated by concepts and ideas from the fields of epidemiology and evolutionary biology. The community is generated from sequences of successful "contacts'' between various types of individuals, which generate multiple nonlinearities in the corresponding differential equations that form the model. A bifurcation analysis of the model provides an explanation for the impact of individual learning on community intellectual development, and for the resilience of communities constructed via multilevel epidemiological contact processes, which can survive even under conditions that would not allow them to arise. This simple cooperative framework thus addresses the generalized belief that sharp community thresholds characterize separate learning cultures. Finally, we provide an example of an application of the model. The example is autobiographical as we are members of the population in this "experiment".
Citation: Nicolás M. Crisosto, Christopher M. Kribs-Zaleta, Carlos Castillo-Chávez, Stephen Wirkus. Community resilience in collaborative learning. Discrete & Continuous Dynamical Systems - B, 2010, 14 (1) : 17-40. doi: 10.3934/dcdsb.2010.14.17
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