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Students’ Conceptions of Congruency Through the Use of Dynamic Geometry Software

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Abstract

This paper describes students’ interactions with dynamic diagrams in the context of an American geometry class. Students used the dragging tool and the measuring tool in Cabri Geometry to make mathematical conjectures. The analysis, using the cK¢ model of conceptions, suggests that incorporating technology in mathematics classrooms enabled a measure-preserving conception of congruency with which students’ could shift focus from shapes to properties. Students also interacted with dynamic diagrams in a novel way, which we call the functional mode of interaction with diagrams, relating outputs and inputs that result when dragging a figure. Students’ participation in classroom interactions through discourse and through actions on diagrams provided evidence of learning using tools within dynamic geometry software.

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Notes

  1. Note that we are here describing a conception of congruency that might help explain some behaviors of students when they are determining congruence. We are not asserting that such conception is correct or even adapted to describe all practices related to congruency.

  2. All names are pseudonyms.

  3. In the second period, the teacher recorded in the table students’ results after a brief discussion, and made diagrams to illustrate one of the ten entries of the table. In the seventh period, the teacher made diagrams for two out of the nine entries of the table. In addition, she made diagrams for one case that was not recorded on the table. In those cases, the teacher asked students to look at the diagram and identify the midpoint quadrilateral.

  4. Transcription conventions include descriptive comments in brackets. These describe actions or visual elements on the board.

  5. Also known as Varignon’s theorem.

  6. Also known as the midpoint connector theorem.

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Acknowledgments

Research reported in this article was carried out with the support of NSF CAREER grant REC 0133619 to the second author and while the first author was a doctoral student, under the direction of the second author, at the University of Michigan. While contributing to that project, the first author was supported by a Rackham Fellowship from the University of Michigan. Opinions expressed by the authors do not necessarily represent the views of the National Science Foundation or the University of Michigan. A prior version of this article was presented at the 2007 Annual Meeting of the American Educational Research Association in Chicago. The authors acknowledge valuable feedback from Nicolas Balacheff and two anonymous reviewers.

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Authors and Affiliations

  1. University of Illinois at Urbana-Champaign, Champaign, USA

    Gloriana González

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

    Patricio G. Herbst

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  1. Gloriana González
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  2. Patricio G. Herbst
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Correspondence to Patricio G. Herbst.

Appendix 1: Tables of Properties

The teacher used this table to summarize conjectures gathered by students in the first two days of the unit. We translated symbolic notation to denote congruent angles, parallel sides, and parallelograms to make the table more readable.

Appendix 1: Tables of Properties

See Tables 1 and 2.

Table 1 Properties by students in the 2nd period
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Table 2 Properties by students in the 7th period
Full size table

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González, G., Herbst, P.G. Students’ Conceptions of Congruency Through the Use of Dynamic Geometry Software. Int J Comput Math Learning 14, 153–182 (2009). https://doi.org/10.1007/s10758-009-9152-z

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