# Fine grain assessment of students’ mathematical understanding: participatory and anticipatory stagesin learning a new mathematical conception

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## Abstract

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 > 1/10 although 10 > 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.

## Keywords

Anticipation Activity-effect relationship Assessment Children’s conceptions Constructivist theory Fractions Problem situations Tasks Teaching activities Teaching experiment## Notes

### Acknowledgements

This research was supported by a grant from the US *National Academy of Education-The Spencer Foundation*. The opinions expressed do not necessarily reflect the views of the foundation.

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