Rod Nason (Queensland University of Technology, Australia) and Earl Woodruff (OISE-University of Toronto, Canada)

DOI: 10.4018/978-1-60566-198-8.ch293

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TopOne of the most promising pedagogical advances for online collaborative learning that has emerged in recent years is Scardamalia and Bereiter’s (1996) notion of knowledge-building communities. Unfortunately, establishing and maintaining knowledge-building communities in computer-supported collaborative learning (CSCL) environments such as Knowledge Forum® (Scardamalia & Bereiter, 1998) in the domain of mathematics has been found to be a rather intractable problem (Bereiter, 2002a; Nason, Brett, & Woodruff, 1996). Two major reasons for why computer-supported knowledge-building communities in mathematics have been difficult to establish and maintain have been:

*1.*Inability of most “school” math problems to elicit ongoing discourse and other knowledge-building activity (Lesh & Doerr, 2003; Nason, Brett, Woodruff, 1996).

*2.*Limitations inherent in most computer-based math representational tools (De Corte, Verschaffel, Lowyck, Dhert, & Vanderput, 1999; Nason et al., 1996).

Therefore, in this chapter, we argue that if mathematics education is to exploit the potentially powerful new ways of learning mathematics being provided by online knowledge-building communities, then the following innovations need to be designed and integrated into CSCL environments:

*1.*Authentic mathematical problems that involve students in the production of mathematical models that can be discussed, critiqued and improved, and

*2.*Comprehension modelling tools that: (a) enable students to adequately represent mathematical problems and to translate within and across representation modes during problem solving, and (b) facilitate online student-student and teacher-student hypermedia-mediated discourse.

Both of the above innovations are directed at promoting and sustaining mathematical discourse. The requirement that the mathematical problems need to be authentic ensures that students will have the contextual understanding necessary to promote a discussion about the mathematical models. Comprehension modelling (Woodruff & Nason, 2003) further promotes the discourse by making student understanding an additional object for discussion.

TopMost school math problems do not require multiple cycles of designing, testing and refining (Lesh & Doerr, 2003), and therefore, do not elicit the collaboration between people with different repertoires of knowledge that most authentic math problems elicit (Nason & Woodruff, 2004). Another factor that limits the potential of most school math problems for eliciting knowledge-building discourse is that the answers generated from school math problems do not provide students with much worth discussing (Bereiter, 2002a).

Another factor that has prevented most students from engaging in ongoing discourse and other mathematical knowledge-building activity within CSCL environments is the limitations inherent in most computer-based mathematical representational tools (Nason et al., 1996). Most of these tools are unable to carry out the crucial knowledge-building functions of: 1) generating multiple representations of mathematical concepts, 2) linking the different representations, and 3) transmitting meaning, sense and understanding.

Two clear implications can be derived from this. First is that different types of mathematical problems that have more in common with authentic types of mathematical problems investigated by mathematics practitioners than most existing types of school math problems need to be designed and integrated into CSCL environments. Second, a new generation of iconic computer-based mathematical representation tools also need to be designed and integrated into CSCL environments. To differentiate these tools from previous computer-based iconic math representation tools, we have labeled our new generation of tools as comprehension modelling tools. Each of these two issues will be discussed in the next two sections.

Comprehension Modelling Tools: Math representation tools that enable users to: 1) generate multiple representations of mathematical concepts and processes, 2) dynamically link the different representations, 3) communicate the mathematical ideas they have constructed, and 4) make movie-like sequences of animation slides that enable others to replay the process used to generate the solution.

Knowledge Forum®: A single, communal multimedia database designed to facilitate computer-supported collaborative learning.

Mathematical Representations: Concrete, pictorial and symbolic models used to represent mathematical ideas.

Problem Solving: A situation involving an initial state, a goal (or solution) state and a blockage between the initial and goal states that requires the construction of new knowledge to proceed from the initial to the goal state.

Knowledge-Building: The production and improvement of knowledge objects that can be discussed, tested, compared, hypothetically modified and so forth; and not simply the completion of school tasks.

Model-Eliciting Problems: Mathematical problems that involve producing models for constructing, describing, explaining, manipulating, predicting and controlling complex systems (Lesh & Doerr, 2003).

Computer Supported Collaborative Learning (CSCL): Collaborative learning mediated by computers.

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