Students' mental acts of anticipating in solving problems involving algebraic inequalities and equations

Anticipating is the mental act of conceiving a certain expectation without performing a sequence of detailed operations to arrive at the expectation. This dissertation seeks to characterize students' problem-solving in terms of two types of anticipating acts: (a) foreseeing an action, which refers to the act of conceiving an expectation that leads to an action, prior to performing the operations associated with the action, and (b) predicting a result, which refers to the act of conceiving an expectation for the result of an event without actually performing the operations associated with the event. Harel's (in press) triad of determinants--mental act, ways of understanding, and ways of thinking--is used to analyze students' acts of foreseeing and predicting. This research has three objectives: (a) to categorize students' ways of thinking associated with foreseeing and predicting, (b) to identify the relationships between these ways of thinking and students' ways of understanding inequalities/equations, and (c) to explore the potential for advancing students' ways of thinking associated with foreseeing/predicting. To accomplish these goals, fourteen 11th enrolled in various mathematics courses were interviewed. Four of them participated in one-on-one teaching interventions. Non- directive tasks were used to elicit students' anticipatory behaviors. In this study, five ways of thinking associated with foreseeing were identified: impulsive anticipation, tenacious anticipation, explorative anticipation, analytic anticipation, and interiorized anticipation. Three ways of thinking associated with predicting were identified: association-based prediction, comparison-based prediction, and coordination-based prediction. In addition, five ways of understanding inequalities/equations (I/E) were identified: I/E-as-a-signal-for-procedure, I/E-as-a-static -comparison, I/E-as-a-proposition, I/E-as-a-constraint, and I/E-as-a-comparison-of-functions. Students' ways of thinking associated with foreseeing/predicting were found to be related to the quality of their solutions as well as the sophistication of their ways of understanding inequalities/equations. One learner's improvement was summarized in terms of the change in the sub-context (Cobb, 1985) in which she operated, from manipulating symbols in the pre interview to reasoning with symbols in the post- interview. Her operating in the sub-context of working with numbers helped her to achieve this transition. This finding underscores the importance of using numbers as a platform for algebra students to explore algebraic expressions and symbolic structure