Future directions and perspectives for problem solving research and curriculum development

The topic of the present research belongs to the educational studies of problem solving. The general goal of the current article is to suggest new perspective to teach mathematical problem solving through reviewing history of research on that topic and revealing several existing limitations.

However history of research on mathematical problem solving begun 60 years ago, according to the literature review there were revealed fairly ambiguous results about optimal teaching practices towards problem solving. From one point of view it is better to teach specific heuristics or strategies to solve problems, and from another point of view it is more rational to develop the whole concept understanding (or develop metacognitive strategies). These two traditions of studies are called descriptive and prescriptive, respectively. And the problem is the first approach of teaching focuses on strategies without understanding the whole concept and the second approach of teaching suggests apply too general and for this reason unclear rules to problem solving.  In other words it is still unclear whether the concept development or the development of competencies of problems solving is the best way to learn problem solving. 

Taking into account limitations of previous studies authors suggest Models&Modelling perspective as an alternative way to teach problem solving. Instead of describing the external characteristic of problem solvers’ behavior, authors propose to look deep inside the process and try to understand the nature of this process. In other words they are more interesting in such steps of modelling process as mathematizing problems, quantifying a situation, interpreting results, etc. Moreover, according to their concept the M&M approach helps learners consider the problem as a whole process rather than parsing a problem in separate pieces.

From my opinion the Models&Modeling is the most optimal approach to learn problem solving, however there are still remained many unanswered question about teaching practices. For example, whether we should start with analyzing examples of solved problems, or it is better to start solve problem without knowing how to do it, or it is better to teach solve problem step by step according steps in modelling process, etc. Moreover, it is possible that students could have different problems with acquiring process of modelling. Some of them will struggle at mathematizing step while others will stick at the step of interpreting results.