This study reports on the algebraic generalisation strategies used by two fifth grade students along with the factors that appeared to influence these strategies. These students were examined over 18 instructional sessions using a teaching experiment methodology. The results highlighted the complex factors that appeared to influence student strategy use, which included: (a) input value, (b) mathematical structure of the task, (c) prior strategies, (d) visual image of the situation, and (e) social interactions with the teacher and other student. Particular combinations of these factors appeared to increase the predictability of student strategy use. However, the complex nature of the factors influencing these strategies demonstrates the challenges that exist in encouraging students to move toward more sophisticated strategies.

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In this study we examined how two students viewed the general nature of their proportional reasoning errors as they attempted to generalize numeric situations. Using a teaching experiment methodology we studied the reasoning of two students over 18 instructional sessions. One student, Dallas, appeared to recognize that the proportional reasoning error applied to all cases of a particular problem situation and began to apply this reasoning across problems. The other student, Lloyd, exhibited difficulty seeing the generality of his mistaken use of proportional reasoning and regularly repeated this error during the study. From the data we developed a schematized description of how students view the generality of their errors and tracked the changes in these views over the course of the study. We analyze how Dallas and Lloyd's perception of errors shaped their understanding of proportional reasoning and provide suggestions for the role errors play in restructuring a student's conceptual schema.

Give me a productive error over a boring, mundane, and unproductive fact any day.

Changes to classroom rules of engagement, such as assessment, the curriculum, instruction, and the environment, can produce real results.