Many students have acquired the habit of spontaneously applying algorithms and formulas to solve mathematics problems without analyzing the problems. On the other hand, there are also students who analyze the problem situation and think ahead before they embark on a course of actions. The act of thinking ahead, or anticipating, is valuable in problem solving. In this book, two types of anticipating acts--foreseeing and predicting--are used to characterize the ways students solve problems. The research that led to the categorization of different ways of thinking associated with the act of foreseeing/predicting is reported in this book. These ways of thinking allow educators to communicate the way a student approaches a problem, such as whether the student hastily applies a procedure (impulsive anticipation), is fixated on a particular idea (tenacious anticipation), explores different ideas (explorative anticipation), analyzes the problem situation and identifies a goal (analytic anticipation), or spontaneously applies well-established ways of understanding (interiorized anticipation).

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