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When, how, and why prove theorems? A methodology for studying the perspective of geometry teachers

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Abstract

While every theorem has a proof in mathematics, in US geometry classrooms not every theorem is proved. How can one explain the practitioner’s perspective on which theorems deserve proof? Toward providing an account of the practical rationality with which practitioners handle the norm that every theorem has a proof we have designed a methodology that relies on representing classroom instruction using animations. We use those animations to trigger commentary from experienced practitioners. In this article we illustrate how we model instructional situations as systems of norms and how we create animated stories that represent a situation. We show how the study of those stories as prototypes of a basic model can help anticipate the response from practitioners as well as suggest issues to be considered in improving a model.

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Notes

  1. Comparable study of practical rationality in algebra has been done by Chazan (see Chazan & Lueke, 2008). This work has also touched on the role of justification and proof.

  2. Additionally, as reported by Herbst and Nachlieli (2007) and Weiss and Herbst (2007), we had made hypotheses about the register in which a theorem is formulated and about the need for a theorem to be named. These two hypotheses were particularly relevant in the production of Story 2 and are not discussed in this comparative analysis.

  3. The homework problem is stated as “Given \( \overline{{{\text{AB}}}} \cong \overline{{{\text{AC}}}} ; \) Prove: \( \angle B \cong \angle C \)” and given next to a triangle whose vertices are labeled A, B, and C.

  4. Conversely the task might be stated as a question, such as “if in a triangle two sides were congruent, what else could be concluded?” Students of geometry are rarely held accountable for producing a proof in response to questions like that one.

  5. The first proof exercise dealt with an isosceles triangle ABC with BC parallel to the floor and AB congruent to AC. The second proof exercise deals with another isosceles triangle (drawn in a different region of the board), also labeled ABC but this time having AC parallel to the floor, and \( \overline{{AB}} \cong \overline{{CB}} . \)

  6. In the session where the Story 5 was presented, alternative proofs of the theorems were also suggested. One of them included a proof of the concurrency of medians using coordinates and the equation of a line.

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Acknowledgments

The research reported in this article is supported by NSF grant ESI-0353285 to the first author. Opinions expressed here are the sole responsibility of the authors and do not necessarily reflect the views of the Foundation. The authors acknowledge valuable conversations with Daniel Chazan and with members of the GRIP (Geometry Reasoning and Instructional Practices) research group at the University of Michigan, in particular, valuable suggestions from Michael Weiss.

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  1. University of Michigan, Ann Arbor, MI, USA

    Patricio Herbst & Takeshi Miyakawa

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  1. Patricio Herbst
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  2. Takeshi Miyakawa
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Correspondence to Patricio Herbst.

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Herbst, P., Miyakawa, T. When, how, and why prove theorems? A methodology for studying the perspective of geometry teachers. ZDM Mathematics Education 40, 469–486 (2008). https://doi.org/10.1007/s11858-008-0082-3

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