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The best of both worlds...
In the mathematics classes I attended and instructed, we were more in line with cognitive information processing theory as we taught students to store and retrieve mathematical facts and principles through repetition. Although I believe prerequisite skills are important to the students' ability to solve more complex problems, I am intrigued by the notion that "knowledge remains inert and unused if taught in contexts that separate knowing from doing" (Driscoll, 2000, p.155). The example cited by Driscoll from a 1988 article by A. H. Schoenfeld, involved a mathematics problem where students had to determine how many buses, each with a capacity of 36 soldiers, would be needed to transport 1,128 soldiers. Although most of the students were able to correctly solve the long division required to answer the question, approximately one third of those students did not provide the correct answer. Instead of indicating the number of buses that would be required, they simply gave the results of their division problem including the remainder.
Mathematics curriculum developed using the principles of situated cognition where as Driscoll (2000) describes, "knowledge accrues through the lived practices of the people in a society" (p.156) emphasizes the application of knowledge to address problems encountered in real-life situations. This approach to problem solving can provide incentives for students to acquire the necessary skills and knowledge needed to solve the problems, developing them as knowledge workers. Anchored instruction is an interesting application of this philosophy. In 1990, the Cognition and Technology Group at Vanderbilt (CTGV) presented The Jasper Woodbury Series, a set of video disks with stories that presented real-life problem situations to be solved working together in groups with guidance from the teacher (Driscoll, 2000, p. 174).
To develop students who understand the application of mathematics to real-world situations and who know how to collaborate with others to achieve collective goals, I believe that curriculum must create learning environments of situated cognition. Teachers must be facilitators of learning rather than transmitters of facts. In addition, the curriculum must also incorporate the constructivist philosophy of focusing on "higher-order goals" while providing scaffolding to support learners in developing or constructing the necessary fundamentals. This is where the principles of cognitive information processing and the idea of practice and repetition to develop the automaticity of basic skills is a necessity. Integrating this practice within the framework of achieving the collective goal will enable students to understand the value and application of this knowledge.