This study addressed a twofold problem--the soundness of a theoretical stage-distinction regarding the process of constructing a new (to the learner) mathematical conception and how such distinction contributes to fine grain assessment of students' mathematical understandings. As a context for the study served the difficult-to-grasp concept of "inverse" order relationship among unit fractions, that is, the larger the number of parts the smaller the size of each part (e.g., 1/7 greater than 1/10 although 10 greater than 7). I conducted this study as a whole-class teaching experiment in a third grade classroom at a public school in Israel. The qualitative analysis of tasks presented to students and students' responses to those tasks, as well as a quantitative measurement of percents of student responses to assessment questions, indicated that the distinction between a participatory and an anticipatory stage is sound and useful in guiding the teacher's selection of tasks to assess/teach students' mathematical thinking. In particular, this analysis demonstrates that in a classroom where the vast majority of students "appear" to understand a new concept, a substantial portion of the class--those who formed the new conception only at the participatory stage--may be at risk of being left behind. This study also highlights a new way of organizing assessment to minimize such unfortunate situations, including three levels of assessment rigor a teacher can use in regular classroom settings.