The purpose of this paper is to describe (a) multivariable calculus students' meanings for the domain and range of single and multivariable functions and (b) how they generalize their meanings for domain and range from single-variable to multivariable functions. We first describe how students think about domain and range of multivariable functions…

The idea that learning generalizes beyond the conditions of initial learning serves as a basis for our educational system (National Research Council, 2000). That is, educators hope students will use the learning that is generated in the classroom to productively reason about situations they have yet to encounter. One body of research that has…

The concern of the experiment is to find out the roles of abstraction and generalization in the learning of mathematical structures. The basic question is whether to generalize before abstracting or vice-versa in order to maximize transfer. The experiment involves four mathematical tasks and a transfer of activity. Experimental procedures are…

The effects of negative instances on the acquisition of the mathematical concepts of commutativity and associativity were examined. Also investigated were possible transfer effects that might result from the use of negative instances. For 64 ninth-grade subjects, results favored the treatments containing mixed instances and supported the transfer…

The broad goal of this report is to describe a form of knowing and a way of participating in mathematics learning that contribute to and further alternative views of transfer of learning. We selected an episode with an undergraduate student engaged in a number of different tasks involving a physical tool called "water wheel". The embodied…

Junior high school students (N=104) were given instruction using programed booklets on modular addition. Twelve conditions of verbal mediation were randomly assigned to students. Conditions were defined by presence or absence of introductory discussion (2 levels), number of worked examples (2 levels), and type of rule (3 levels: specific, general,…