The purpose of a series of our studies is to investigate the role of "self-referential-activity" in mathematical problem solving. The term "self-referential-activity" means solver's activities that he/she refers to his/her own solving processes or products during or after problem solving. In study (I), we proposed the theoretical framework for analyzing self-referential activity. And, in study (II) and (III), we elaborated the variable "OG/NOG" and "M-SE/SE-C" respectively. In these series of our studies, we have investigated self-referential-activity focusing on the problem-solving phases until the end of problem solving. In this article, we theoretically examined the role of "self-referential-activity", especially "looking-back" activity in the phase after problem solving. And we conducted an experiment to investigate "looking-back" activity after problem solving. At first, we identified six roles of "looking-back" activity. These are as follows: Checking answer or solution, exploring different or more sophisticated solutions, understanding the problem deeply, expanding or elaborating the solution and applying the solution to other problems, thinking critically or creatively, and considering validity or generality of the solution. Secondly, we conducted an experiment to investigate "looking-back" activity after problem solving. Junior high school students (2nd grade, N=117) were divided into two groups. One group was given some questions to encourage solvers to look back their solution of a problem and think other approaches, and another group was not given such the type of question. Subsequently, all subjects were asked to solve more difficult problem. As a result, we found that various "looking-back" activities occurred by simple instructions and these "looking-back" activities were effective to improve solvers' solution. These finding supports our theoretical assumptions and the value and possibility of studying the kind of "self-referential-activities".
