In this chapter I present some of the results of a study aiming to learn if and how using an assessment tool such as a grid of criteria, can be useful for summative assessment and encourage formative assessment processes in the case of inquiry-based mathematics education. I focus my research on a course centred on prob-lem solving in the canton of Geneva (Switzerland). In the first part, I provide a synthesis of teachers' points of view about this course relative to problems they submitted to students, research narrative chosen as a means to assess students and the assessment of students' problem solving competencies. I then describe how it leads to a collaborative work that aims to elaborate a grid of criteria of re-search narratives to assess students' problem solving competencies with both a summative and formative purpose. Finally, I describe an exploratory study; an analysis of two lessons taught by a teacher who uses the grid of criteria, in order to understand if and how she refers to these criteria to develop informal formative assessment.