2015 Volume 21 Issue 2 Pages 133-145
The purpose of this paper is to develop a strategy for designing mathematical tasks to motivate the shift to the mathematical problem solving phase, that is, the phase of understanding mathematical problems. This paper mainly consists of five parts. First, we review radical constructivism as a theoretical background. In order to focus on whether each student can understand mathematical problems, we develop the strategy from the radical constructivist point of view. Second, we formalize the central concepts of radical constructivism. This formalization makes us easily use those concepts for designing mathematical tasks. Third, we formalize the meaning of mathematical problems for students. From the radical constructivist point of view, there can be a gap between the students’ interpretations of mathematical problems and those of the teacher. We discuss what conditions make mathematical tasks easily understandable for students. Forth, through a consideration on “a problem of determining the minimum value of quadratic functions, where the equations of their vertical lines include constants” within a Japanese high school mathematics textbook, we propose a strategy for designing mathematical tasks to motivate the shift to the mathematical problem solving phase. Concretely, the strategy consists of the following procedures for improving the way of posing mathematical tasks: [1] To make students confirming two important conceptual frameworks about the search range of the answer and about the criterion of validity of the answer; [2] To make students experience perturbation when they apply their alreadyknown knowledge-how; [3] To make students experience appropriate accommodation of the conceptual framework for assimilating mathematical problems. This proposal is based on the application of the conditions to make mathematical tasks understandable for students. Finally, we discuss the implication from our proposal to future teaching and researching practices.
