The role of generic examples in teachers’ proving activities

Abstract

This paper explores how in-service teachers enrolled in a graduate proof course interpret, understand, and use generic examples as part of their proving and justification activities. Generic examples, which are capable of proving and justifying with strong explanatory power, are particularly important for teachers considering teaching proof in their classrooms. The teachers in our study used generic examples to produce three types of proof: example-based arguments enhanced with generic language; incomplete generic examples; and complete generic examples. We found that teachers conflate generic examples and visual representations, prefer visual generic examples for teaching, and consider a generic example with symbolic representation to be more convincing than a generic example without. We conclude with implications for secondary school teaching, as well as suggestions for future professional development efforts.

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Notes

  1. 1.

    This research was conducted in the USA, where K-12 refers to formal education beginning with early childhood (Kindergarten) and concludes with 12th grade (when students are generally around 18 years of age). 12th grade is the last grade before college.

  2. 2.

    Although there is an ongoing debate as to whether a generic example can be used to fully prove a theorem (e.g., Balacheff, 1988; Harel & Sowder, 1998; Leron & Zaslavsky, 2013; Rowland, 2002), our focus is on how a classroom community uses, perceives, and learns from generic examples in their proving activities. Consequently, this debate, important though it is, is beyond the scope of this paper.

  3. 3.

    In the analysis, we did not separate GEs produced before and after the instructor introduced the terminology, because we found no difference before and after.

  4. 4.

    One important note is that individual teachers would present their group’s work to the class as a whole, and consequently it will sometimes appear in the data that one teacher is presenting her own work, but in actuality, she is presenting her entire group’s work.

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Acknowledgments

Thanks to Eric Knuth, Amy B. Ellis, Percival G. Matthews, Ana Stephens, and Rosemary Russ, the dissertation committee, that guided this data collection with the first author. Thanks to Jordan T. Thevenow-Harrison for their repeated reviews of this manuscript throughout the process, to Elise Lockwood for their helpful comments on an earlier version of this manuscript, and to the editor and the anonymous reviewers for their immensely helpful feedback. We are deeply grateful to the instructor who shared her classroom and her expertise and the teachers who shared their learning.

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Correspondence to Muhammed Fatih Dogan.

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Dogan, M.F., Williams-Pierce, C. The role of generic examples in teachers’ proving activities. Educ Stud Math (2020). https://doi.org/10.1007/s10649-020-10002-3

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Keywords

  • Generic examples
  • Proof and justification
  • Teacher education
  • Proving
  • Exemplifying
  • Visual and algebraic representations