linguistic and socio-cultural diversity – is seen as the “engine” that drives the emergence and development of
content in this group-oriented approach to generativity (e.g., see Ares & Stroup in this volume). Diversity creates
the space of possibilities that students and teachers can use to advance teaching and learning.
The group focus of next-generation classroom networks develops from, and is consistent with, this expanded
sense of generativity. These systems are typically designed “from the ground up” with the classroom in mind.
Rather than constraining the learning experience to be somewhat narrowly individualistic, these technologies
support socially situated interaction and investigation. Typically, each student has a device that allows him or her to
participate either synchronously or asynchronously in group-oriented activities. Often the devices have local input,
display, and analysis capabilities (e.g., those of a graphing calculator) and can interact in the network with each
other and even with other kinds of devices (e.g., data-collection tools like Calculator-based Rangers
TM
motion
detectors as used in the People-Molecules activity developed by Wilensky and Stroup [in review]). The interactions
and emergent results can be projected on a public display space – often in real-time – using a computer or calculator
projection system. Using this infrastructure and generative design, the learning trajectories and the processes of
knowledge construction are owned by the group itself. Software and hardware work together in supporting this
“group-oriented” design. The classroom or group, as supported by the network infrastructure, becomes a kind of
participatory manipulative for teaching and learning challenging mathematical and/or scientific content.
A significant number of these networks are about to become widely available and are poised to become a major
presence in classroom learning. It behooves us, then, to begin to support approaches to optimizing the generative
potential of these networks. Although many of the forms and kinds of activities discussed using the pathways and
endpoints discussion below can be done without next generation network technology and, indeed, have been used in
pre-service teacher education (cf., Stroup, 1997), the sense is that the highly interactive, and group-focused
capabilities of these next-generation systems can support, extend, and add to this kind of activity design for
classrooms.
3.0 What are Generative Activities?
Generative activities in group contexts are seen as space creating activities that extend the ideas of co-operation
and emergent structure to classroom based activity. Learners create a space – or coordinated collection – of
expressive artifacts and actions in relation to some shared task or set of rules. The structures that are created or
embodied are not determined in advance but are co-constructed by learners as their sense-making evolves and
develops. Students can be asked to create objects or outcomes that are the same or alike in some mathematically or
scientifically significant sense (see taxonomy below) and these responses can be arrived at or built from the
underlying sameness in a wide range of ways. The sameness gives coherence to the task. Generativity requires that
the activity produce or give origin to a diversity of responses. This diversity is then used by the group to explore
patterns in the responses and in the structural ways in which the responses might be seen as relating to one another
or of-a-whole. Ideally the space of responses will be large enough so that the kinds of behaviors or expressive
artifacts students create give the teacher significant insights into the ways the students are thinking about the task.
The activity should be “thought revealing” in this sense (Lesh, et al., 2000), and it should also be capable of giving
rise to additional rounds of generative exploration and/or detailed investigation.
4.0 Taxonomy of Generative Activities
The following taxonomy of generative activities is organized in terms of pathways and endpoints. Pathways are
intellectual and/or behavioral routes for arriving at a given endpoint. Endpoints are outcomes or artifacts created by
learners that represent some form of completion of the given generative task (See Figure 1.).
pathways
endpoints
Figure 1. The taxonomy centers on a pathways and endpoints analysis of generativity.
4.1 Nominally Generative
Convention