College students' understanding of variable in a traditional, developmental, Elementary Algebra curriculum
Reisch, Christopher Paul
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Variable is the central concept of the traditional beginning algebra curriculum at two-year colleges, particularly when used to represent an unknown number, a general number, or to express a functional relationship. This study set out to determine the degree to which beginning algebra students can work with variables in these three roles, what errors they make and what strategies they use when doing so, and whether or not there is a cognitive hierarchy to the different categories of variable use. A 38-item questionnaire was given to 146 Elementary Algebra students from a medium sized, public, two-year, community college in western New York State. Responses were summarized, categorized, and compared to results of previous research with other groups of students, and then subjected to Guttman scalogram analysis in order to assess the scalability of the categories of variable. The group of students under study in this investigation displayed many of the same approaches, misunderstandings, and difficulties that have been reported over the past several decades. They were generally facile in dealing with variable as unknown items when the complexity of the item was low to moderate, but as soon as the complexity increased, performance declined substantially. Students had some success dealing with variables as general numbers, but had considerable difficulties coordinating the relationship between two variables. In instances where they were somewhat successful, students took a numeric, substitute and check approach that focused on whole numbers. Students had substantial difficulties in translating written problems into mathematical statements, and often took operational approaches to questionnaire items, even in items where a structural approach was more efficient. Scalogram analysis suggested that the categories of variable were strongly scalable for this group of students. Findings suggest that the traditional beginning algebra curriculum places too much emphasis on basic equation solving procedures at the expense of making generalizations and investigating relationships between variables. In order to better prepare students for the study of advanced mathematical topics, the traditional beginning algebra curriculum and textbooks need to be refocused to make the different uses of variable more explicit and to address these deficiencies.