## Abstract

Decades of research have established that solving geometry proof problems is a challenging endeavor for many students. Consequently, researchers have called for investigations that explore which aspects of proving in geometry are difficult and why this is the case. Here, results from a set of 20 interviews with students who were taught proof in school geometry are reported. Students who earned A or B course-grades in the proof unit(s) were asked to share their thinking aloud while solving two proof tasks using smartpens. Student thinking was analyzed for two subgroups—students who were successful with both proofs (*n* = 7) and who were unsuccessful with both proofs (*n* = 13). Large differences were observed in how often students in the two groups exhibited certain competencies and behaviors. The largest gaps occurred in the ways in which students attended to the proof assumptions, attended to warrants in their proofs, and demonstrated logical reasoning.

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## Notes

Our interpretation of this claim is not that there are some countries teaching proof well, but, rather, that not enough documentary evidence to support a claim that teaching proof is a failure in

*all*countries exists.A small number of data glitches occurred with the smartpen technology.

This is useful information for viewing figures in the Findings.

Rubrics were included in Senk’s (1983) dissertation.

This hypothesis was confirmed when we coded these data later on. We chose not to include these details because they over-complicate the reporting of the findings and were not very enlightening.

P14 refers to the 14th participant. P14–P20 were the SPs, and P1–P13 were the NPs.

We “cleaned” up the transcripts slightly by removing “ums,” “uhs,” and so forth, for ease of reading.

Due to a technology glitch, P10’s markings in this diagram are off-center but were sensibly placed on the paper.

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## Acknowledgements

The authors would like to thank the teachers and their students for allowing them to conduct this study. They also thank Amanda Seiwell and Kevin Felice for supporting data analysis, as well as Keith Weber for useful suggestions on methods and Ron Gallimore for helpful suggestions on the manuscript. The research reported in this paper was supported with funding from the National Science Foundation (NSF; Award #1453493, PI: Michelle Cirillo). Any opinions, findings, and conclusions or recommendations expressed in this material are those of the authors and do not necessarily reflect the views of the NSF.

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Cirillo, M., Hummer, J. Competencies and behaviors observed when students solve geometry proof problems: an interview study with smartpen technology.
*ZDM Mathematics Education* **53**, 861–875 (2021). https://doi.org/10.1007/s11858-021-01221-w

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DOI: https://doi.org/10.1007/s11858-021-01221-w