2018 Volume 24 Issue 1 Pages 135-145
The aims of this paper are to propose strategies for supporting mathematical problem solving from a new perspective of academic abilities of mathematics and to build hypotheses regarding what opportunities prompt learners to apply mathematical methods in mathematical problem solving. Since hypotheses should be falsifiable, it is important for our scientific endeavor to simultaneously achieve these two aims. This paper will provide a starting point for the cycle of theory building and practical verification. For this, we will analyze a conventional high school mathematical problem and conjecture what supports learners to shift their attention to a key aspect of the problem from a radical constructivist point of view.
As a result, we will build the following two hypotheses about opportunities for applying mathematical methods along with two corresponding strategies for supporting learners to obtain such opportunities. First, we will propose that an opportunity for applying mathematical methods dependent on particular mathematical content is to establish a cognitive status of noticing the effectiveness of the methods in the current context without proving it. In order to provide such an opportunity, it can be effective to encourage learners to seek instances satisfying the given conditions. Second, we will hypothesize that an opportunity for applying mathematical methods independent from particular mathematical content is to understand the cultural nature of mathematical activities. In order to provide such an opportunity, it can be effective to familiarize students with customs in mathematics in an explicit manner. It is concluded that dependence on mathematical content is a key property of designing an approach for teaching mathematical methods. The distinction between different mathematical methods should be made in new discourses adopting a new perspective of academic abilities of mathematics.
