What makes a problem mathematically interesting?

Sandra Crespo and Nathalie Sinclair poses this very interesting question in an article that has recently been published in Journal of Mathematics Teacher Education. The entire title of the article is: What makes a problem mathematically interesting? Inviting prospective teachers to pose better problems.

Mathematical problems are an integral part of mathematical learning, and although most pupils encounter mathematical problems as they are posed in textbooks, the teachers have an important role in assigning appropriate problems for the students to solve. Prospective teachers have had few opportunities to focus on problem posing in their studies, and their experience with mathematical problems are mostly in connection with the solving of problems that are posed by the teacher or a textbook. The authors of this article "consider the practice of problem posing to be especially important for prospective teachers because a great deal of the work of teaching entails the posing and generation of what the mathematics education community often refers to as “good” questions—questions that aim to support students’ mathematical work".

The main research questions in the study described in this article are:

1. What is the role of exploration in the problem-posing process? (What happens when prospective teachers pose problems with and without first exploring the situation that could motivate their questions? What kinds of questions do they pose in each of these two kinds of structured problem-posing setting?)
2. How do prospective elementary teachers decide on the quality of the questions they pose? (What rationale do they provide when asked to justify what makes their questions mathematically interesting? What is the effect of making explicit some of the qualities that make mathematics problems interesting and worth solving?)

The questions were investigated in a course that Sandra Crespo taught herself, and the course was offered in the fourth year of a 5-year teacher preparation program. A central theme in the course was a "pedagogy of inquiry" rather than one of presentation, and the students were given the opportunity to investigate different forms of mathematics teaching. There were 22 students in the course, and the researchers used four tasks and two classroom interventions in the study. The data consisted of written work from the students as well as field notes from observations of the students' work with the given tasks, and from discussions in class.

Here is the abstract:

School students of all ages, including those who subsequently become teachers, have limited experience posing their own mathematical problems. Yet problem posing, both as an act of mathematical inquiry and of mathematics teaching, is part of the mathematics education reform vision that seeks to promote mathematics as an worthy intellectual activity. In this study, the authors explored the problem-posing behavior of elementary prospective teachers, which entailed analyzing the kinds of problems they posed as a result of two interventions. The interventions were designed to probe the effects of (a) exploration of a mathematical situation as a precursor to mathematical problem posing, and (b) development of aesthetic criteria to judge the mathematical quality of the problems posed. Results show that both interventions led to improved problem posing and mathematically richer understandings of what makes a problem ‘good.’