The structure and function of the cognitive constructs engaged during the symbolization of linear situations
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Providing experiences that allow students to symbolize the relations embedded in situations is an important aspect of mathematics education. Research has shown that knowledge from several domains is incorporated into symbolization processes and that understanding of the situation, both the problem statement and the physical situation that encompasses a relation, is a key component in the representation process. This work builds on that research by describing the structures and functions of the cognitive constructs that students engage as they generate equations for linear relations embedded in common situations. A schema model is developed to describe the structure and functions of the cognitive constructs employed by College Algebra students as they attempt to construct equations for linear relations during task based interviews. The responses of four students are analyzed in an attempt to describe the objects and relations that comprise their constructs, the role of the situation schema in symbolization, and the connections between knowledge areas. Regardless of their level of ability, students combined similar objects in their constructs. Differences appear in how they related the objects and mapped both objects and relations between the different knowledge areas. Students appeared to organize task information into a format that is congruent with frames in other knowledge areas. This suggests that the flow of information is more multi-directional than some of the previous research has implied. The ability to symbolize was mirrored by the degree to which relational aspects were considered, incorporated, and maintained in the representations. Connections between knowledge areas appear to occur through shared nodes and mappings that connect different presentation forms of the objects and relations.