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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.

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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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  1. Beering Hall of Liberal Arts and Education (Room 4122), Department of Curriculum and Instruction, Purdue University, 100 N. University St., West Lafayette, IN, 47907-2098, USA

    Ron Tzur

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  1. Ron Tzur
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Correspondence to Ron Tzur.

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On December 28, 2004, as the author was about to submit the manuscript of this paper for review, he was notified that his daughter, Shiri, was admitted to the emergency room of a local hospital. Shiri died on January 5, 2005, at the age of 22, after struggling for 9 days with brain damage caused by medical malpractice. This paper is dedicated in her memory.

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Tzur, R. Fine grain assessment of students’ mathematical understanding: participatory and anticipatory stagesin learning a new mathematical conception. Educ Stud Math 66, 273–291 (2007). https://doi.org/10.1007/s10649-007-9082-4

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