From Jsarmi

Here are some reflections tracing the notion of a "problem space" and how it" can be considered to be a continuous (although contingent) achievement of a group engaged in collaborative problem solving

Contents 1 Problem Space 2 Joint/Shared Problem Space 3 Group Cognition 4 Boundaries of a Shared Problem Space 5 Groups and their Shared Problem Space 6 References

Problem Space

The idea of a problem space is central to the information-processing or cognitive view of human problem solving (i.e. Newell and Simon's problem-solving theory). From this perspective, the task environment gets abstracted into an internal representation which is what the problem solver uses (i.e. manipulates symbolically) to actually solve the problem. This internal representation is what a subject is assumed to construct when s(he) understands a task correctly and might contain (encode):

• The current initial state of the problem;
• The goal state or condition (as a test procedure to know when one is done, or a declaration of what the "end" is)
• Constraints determining allowable moves and states (e.g. nodes and allowable links of the space)
• Other representations encoding knowledge of problem solving 'methods, heuristics, or metrics specific to the current task environment, ideas about when one is getting closer to a solution, common ways of overcoming stuck states, etc.

This conception of problem-solving as search through a problem space "localized" in the head of an individual was specially insufficient when analyzing problem-solving in everyday life. For example, when Lave studied mathematical activity in supermarkets she observed how such activity was centrally rooted in "being “in” the “store”, walking up and down “aisles”, looking at “shelves” full of cans, bottles, packages and jars of food, and other commodities." Many other similar investigations challenged the restrictive notion of what should be taking into consideration when speaking of a "problem space." Some of these challenges also came from investigations of collaborative problem-solving and the use of artifacts during problem-solving activity.

Joint/Shared Problem Space

Roschelle and Teasley proposed the notion of a joint problem space (JPS) as a "shared knowledge structure" that supports problem solving activity in collaborative contexts by integrating:

• Goals
• Descriptions of the current problem state
• Awareness of available problem solving actions

Initially, this description does not seem too radically different than the traditional cognitive view. After all, one can conceive each one of these elements as internal representations and the "shared knowledge structure" as the intersection of these individual "mental models" or individual problem spaces. In fact, this has been the predominant psychological interpretation of collective cognition (e.g. Fiore and Salas' theory of team cognition'). However, Roschelle and Teasley's proposal goes beyond this simple interpretation. From their perspective "social interactions in the context of problem-solving activity occur in relation to a "shared conception of the problem" which results from a collaborative process of coordination of communication, action, and representation in a particular context of activity, not in the heads of the individuals. This perspective, and indeed, their method of analysis, are consistent with Garfinkel's ethonmethodological observation about the nature of "shared agreements:"

"Shared agreement" refers to various social methods for accomplishing the member's recognition that something was said-according-to-a-rule and not the demonstrable matching of substantive matters. The appropriate message of a common understanding is therefore an operation rather than a common intersection of overlapping sets. (p.30)

From this perspective, a "shared agreement" or a "mutual conception of the problem" is then the emergent and situated result of the participant's interactions, embodied in their context of activity. In the words of Roschelle and Teasley, it is "the coordinated production of talk and action by two participants (what) enabled this construction and maintenance (of the joint problem space) to succeed... the introduction of successful ideas was sometimes asymmetric, although it succeeded only through coordinated action." One analytical problem resulting from this perspective is that of situating the three elements of the joint problem space (goals, current problem state, and awareness of available actions) in interaction and linking such interactional products with the cognitive reality of individuals.

More recently, other researchers have furthered the notion of a joint problem space adding more evidence to the importance for social interaction in the construction, use and maintenance of such resources. Baron (2003), for instance, argued beyond the need to "manage joint attention at solution-critical moments" as explored by Roschelle, groups face other complex challenges that when successfully managed accounted for their overall advantage over groups with equivalent capacity. In particular, she argued that there are critical challenges to the creation of joint problem-solving spaces as individual participants are simultaneously managing their own efforts to understand pieces of the problem and at the same time trying to understand what others are doing. As a result, Baron argues that "collaboration might productively be thought of as involving a dual-problem space that participants must simultaneously attend to and develop." This dual space" comprises:

• A content space consisting of the problem to be solved, and
• A relational space consisting of the interactional challenges and opportunities.

Baron argues that the content space and relational space are negotiated simultaneously but can compete for limited attention, "one needs to be able to monitor and evaluate one’s own epistemic process while tracking and evaluating others’ epistemic processes." An example of how these two spaces intersect and one that Baron follows through her entire analysis of group problem solving is the management of problem-solving proposals. A problem-solving proposal has both a relationship with the content space or problem space and to the social interaction of the group. It involves elements of the problem or of a strategy that might require attention, offer a new insight, or might indicate a solution path. At the same time, they are "bids" for collective action, requests for co-participants to, perhaps, interrupt their current activity, and in some cases, contrasting alternatives to other proposals already stated. This "dual" character of the interaction space expands our understanding of what a "joint problem space" entails, although, it still remains as a challenge to study their interdependence in-depth.

Group Cognition

The dialectical relationship between social interaction and the construction of meaning (i.e. a problem space) has been directly addressed by the theory of group cognition (Stahl, 2006a). From this perspective, the organization of action is an emergent property of moment-by-moment interactions among actors, and between actors and the activity system in which they participate collectively. The content space' and the relational space are mutually constitutive from this perspective. Stahl, offers a candidate description for how the dynamic process of building knowledge might intertwine the content and relational spaces:

"Small groups are the engines of knowledge building. The knowing that groups build up in manifold forms is what becomes internalized by their members as individual learning and externalized in their communities as certifiable knowledge. At least, that is a central premise of this book." (Stahl, 2006, p. 16)

Stahl, in his analysis of how groups "sustain" their group cognition while engaged in online mathematical problem solving, has alluded to two dimensions of how time might be an important element of problem-solving activity. On the one hand, the collaborative activity involved in solving a problem can be "spread across" hundreds of micro-level interactions. On the other hand, individuals might internalize the meaning co-constructed through interactions and "sustain" the group cognition by engaging in later individual or group work. In either case, "groups sustain their () social and intellectual work by building longer sequences of math proposals, other adjecency pairs and a variety of interaction methods." From this perspective, different aspects of the reality of groups solving problems are illuminated. The following concepts attempt that goal:

The problem-as-presented and the problem-to-be-solved. Initially, a problem might be a set of textual, graphical and other resources available for a group to consider. These resources are used in interaction, and through interaction become a problem-to-be-solved involving a group's own "problematization" (Koschmann et al, 2003) of individual parts of the problem-as-presented or its entirety. This activity can be re-visited multiple times during the interaction with the problem-to-be-solved evolving or changing. This is what in traditional theory of problem-solving was referred as "framing" but was rarely studied in any direct way.

Proposals and courses-of-action. Problematization defines a focus of attention and establishes an initial set of relevancies but it is through exploration, the development of proposals and the interactive engagement with them that problem-solving evolves. Describe here how proposals can become shared courses of action contributing to the development of meaning-objects (e.g. math objects) and collectivities. Explore how proposal work monitors the proposed activity and the actions-according-to-plan ala Suchman

Meaning-objects. math objects, observations, constraints, etc. discovered and stated in interaction to shape the context of activity

Solution objects. objects constructed and treated as solutions whether they actually "solve" the problem in an abstract sense or not

Collectivities-in-action Alignments, orientations, participation frameworks, ephemeral division of labor, etc.

Trajectories-of-action how to start or end something, Awareness of where the group has been, where it could go next, where others have been, etc.

Boundaries of a Shared Problem Space

The distinction between the cognitive view of a problem space as an idealized structure in someones head and a problem space as the emergent of people engaged in problem solving activity, can be explored from the perspective of boundaries:

• Where does the problem space reside?
• When does the problem space can be said to no longer exist?

(Note: Distributed Cognition may have a view of a distributed problem space that is also relevant here?)

Groups and their Shared Problem Space

Here is a possible distinction of the work of groups, pased on both their synchronous problem space and their diachronic problem space

References

Barron, B. (2003). When Smart Groups Fail. Journal of Learning Sciences 12(3), 307–359.

Fiore, S. M., & Salas, E. (2004). Why we need team cognition. In E. Salas & S. M. Fiore (Eds.), Team Cognition: Understanding the Factors that Drive Process and Performance, (p. 235-248). Washington, DC: American Psychological Association.

Garfinkel, H. (1967) Studies in Ethnomethodology. Englewood Cliffs, NJ: Prentice- Hall. (Some essential features of common understandings)

Greeno, J. (2006). Learning in Activity. In R. K. Sawyer (Ed.), The Cambridge Handbook of the Learning Sciences (pp. 79-96). Cambridge: Cambridge University Press.

Koschmann, T., Zemel, A., Conlee-Stevens, M., Young, N., Robbs, J., & Barnhart, A. (2003). Problematizing the problem: A single case analysis in a dPBL meeting. In B. Wasson, S. Ludvigsen & U. Hoppe (Eds.), Designing for change in networked learning environments: Proceedings of the international conference on computer support for collaborative learning (CSCL '03) (pp. 37--46). Amsterdam: Kluwer Academic Press.

Lave, J. (1988). Cognition in Practice: Mind, Mathematics and Culture in Everyday Life. Cambridge University Press, Cambridge, UK.

Roschelle, J. & Teasley, S.D. (1995). Construction of shared knowledge in collaborative problem solving. In C. O’Malley (Ed.), Computer-supported collaborative learning. New York: Springer-Verlag.

Stahl, G. (2006a). Group cognition: Computer support for building collaborative knowledge. Cambridge, MA: MIT Press.

Stahl, G. (2006b). Sustaining Group Cognition in a Math Chat Environment. Research & Practice in Technology Enhanced Learning, 1(2), 85-113.