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Article Excerpt
Eliciting high-level mathematics symbolizing and communicating
from students engaged in mathematics communities of practice has been found to be a challenging problem. In this article, we report on a study where 21 grade six female students engaged in model-eliciting problem-solving with collective discourse mediated by Knowledge Forum[R] Computer Supported Collaborative Learning (CSCL) software achieved the kind of progressive knowledge-building activity that until now had not been achieved in CSCL-mediated mathematics communities. During the course of the study, the students engaged in knowledge-building discourse about and iteratively improved their models for ranking the cities of Canada in terms of livability. The success achieved in having the students engage in this knowledge-building activity was attributed to the contexts provided by the model-eliciting math problem and to contexts and scaffolds for knowledge-building discourse provided by Knowledge Forum[R] during the construction and iterative revisions of the math models.
Jl. of Computers in Mathematics and Science Teaching (2003) 22(4), 345-363
BACKGROUND TO THE RESEARCH
A major pursuit of most practitioners working in the discipline of mathematics is the production and improvement of mathematical conceptual artifacts (e.g., mathematical ideas, principles, relationships, theories, interpretations, models, etc.). Mathematical conceptual artifacts are human constructions like other artifacts, except that they are immaterial and instead of serving purposes such as cutting, lifting, and inscribing, they serve purposes such as constructing, explaining, and predicting (Bereiter, 2002; Lesh & Doerr, in press). To produce and improve these mathematical conceptual artifacts, mathematical practitioners often form research communities. These communities collectively pursue greater understanding by engaging in knowledge-building discourse where problems are formulated, posed, and investigated (Halmos, 1980), conjectures are made, and findings are shared and critiqued (Bruce & Easley, n.d.). An essential characteristic of these mathematical problems is that they tend to be open-ended and nontrivial in nature, and typically involve several "modeling cycles." The problems are investigated and, based on feedback from community members, the explanations and predictions are gradually refined, revised, or rejected (Lesh & Doerr).
This process contrasts with the sort of mathematical activity most children engage in within their classrooms. In most mathematics classrooms, the major focus of students is on the completion of tasks such as worksheet and textbook exercises. This focus is not conducive for knowledge-building discourse (Bereiter, 2002). Furthermore, in almost all "school" mathematical problems, students are required to search for an appropriate tool (e.g., operation, strategy) to get from the givens to the goals. The product that students are asked to produce is a definitive response to a question that has already been interpreted by someone else. It can be argued that most classroom discourse about mathematics and mathematical problem solving activity in classrooms tends to be an atrophied version of the discourse engaged in by practitioners working in the living discipline of mathematics (Bereiter, 2002; Lesh & Doerr, in press; Nason, Woodruff, & Lesh, 2002; Schoenfeld, 1992).
This apparent lack of authenticity in mathematical classroom activity has led many seminal thinkers in the field of mathematics education, such as Paul Cobb, Magdalene Lampert, Ed Silver, Richard Lesh, and Helen Doerr, to advocate for major reforms in how mathematics is taught and learned. Many of the reforms place emphasis on the need to change the nature of classroom discourse to include authentic mathematical activity, collaborative mathematical thinking and "talk in the spirit of disciplinary work" (Cobb, 1994; Lampert, Rittenhouse, & Crumbaugh, 1995; Stein, Silver, & Smith, 1998).
Other reforms have focused on the kind of problems and the processes that student work through. This has been the major thrust of the reforms being advocated by Lesh and Doerr (in press) with their seminal work on model-eliciting problem-solving. The design of model-eliciting problems is informed by the following set of principles (Lesh, Hoover, Hole, Kelly, & Post, 1999):
1. The Personal Meaningfulness Principle (sometimes called the reality principle): Is the problem personally meaningful to the students? Does the problem relate to phenomena in which they are intrinsically interested?
2. The Model Construction Principle: Does the task create the need for a model to be constructed, or modified, extended or refined? Is attention focused on underlying patterns and regularities?
3. The Self-Evaluation Principle: Will students be able to judge for themselves when their responses are good enough? For what purposes are the results needed? By whom? When?
4. The Model-Documentation Principle: Will the response require students to explicitly reveal how they are thinking about situation? What kind of system (math objects, patterns, regularities) are they thinking about?
5. The Simple Prototype Principle: Will the solution provide a useful prototype (or metaphor) for interpreting a variety of other structurally similar situations?
6. The Model Generalization Principle: Can the model generated be applied to a broader range of situations? Is the model reusable, sharable, and modifiable?
These principles imply that in model-eliciting problem solving, the products created by students include more than just answers to questions; they involve the production of models or other conceptual artifacts for constructing, describing, explaining, manipulating, predicting and controlling complex systems.
The research being reported in this article coordinates the socio-cognitive discursive approach (advocated by Cobb, Lampert, & Silver) with a model-eliciting problem solving approach (advocated by Lesh & Doerr, in press) in a process that takes advantage of the strengths of each. We argue that adding a collaborative discursive component to Lesh's model-eliciting problem solving approach will enhance the authenticity of classroom mathematical activity. Students will enrich their understanding of mathematical concepts and processes and their understanding about the nature and the discourse of mathematics. We also argue that Lesh's principles, when framed from a collaborative knowledge-building stance, provide boundaries and structure to students' own mathematical discourse.
While it may be possible to integrate a socio-cognitive discursive approach and a model-eliciting problem solving approach in the noncomputerized classroom, we suggest that computer supported collaborative learning (CSCL) environments such as Knowledge Forum[R] have a...
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