Relation between students' behaviors and their mathematical ideas

Lisa B. Warner has written an article that was published online in The Journal of Mathematical Behavior yesterday. The article is entitled How do students’ behaviors relate to the growth of their mathematical ideas? Here is the article abstract:

The purpose of this study is to analyze the relationship between student behaviors and the growth of mathematical ideas (using the Pirie–Kieren model). This analysis was accomplished through a series of case studies, involving middle school students of varying ability levels, who were investigating a combinatorics problem in after-school problem-solving sessions. The results suggest that certain types of student behaviors appear to be associated with the growth of ideas and emerge in specific patterns. More specifically, as understanding grows, there is a general shift from behaviors such as students questioning each other, explaining and using their own and others’ ideas toward behaviors involving the setting up of hypothetical situations, linking of representations and connecting of contexts. Recognizing that certain types of student behaviors tend to emerge in specific layers of the Pirie–Kieren model can be important in helping us to understand the development of mathematical ideas in children.

Warner focus a lot on the Pirie-Kieren model in her theoretical framework (see the article of Susan Pirie and Thomas Kieren from 1994). The main focus of Warner's article is to address the following questions:

Are different types of student behaviors associated with the growth of mathematical ideas in specific ways? If so, how?

In her conclusions, Lisa Warner suggests that for the students in her study, "certain types of behaviors appeared to be associated with the growth of mathematical ideas in certain ways". She also suggests that further research is needed in order to investigate whether these findings correspond with findings in similar studies of other students, different types of tasks, etc.