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High school mathematics teachers’ discernment of invariant properties in a dynamic geometry environment

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Abstract

Variance and invariance are two powerful mathematical ideas to support geometrical and spatial thinking, yet there is limited research about teachers’ knowledge of variance and invariance. In this paper, we examined how high school teachers deal with the task of looking for invariant properties in a dynamic geometry environment (DGE) setting. Specifically, we investigated if they even attend to invariant properties; what invariant properties they discern and discuss; and how DGE can support such discernment. Our analysis found that teachers tend to discern and discuss invariant properties mainly when they were probed to consider invariance. We also found four categories of invariant properties that seem to be important for a robust and rich understanding of geometric objects in the context of invariance and DGE. The use of DGE allowed teachers to see and interact with invariant properties, thus suggesting that accessing geometry dynamically may have structural affordances especially when exploring invariance. Teachers were able to enact different DGE movements to discern and discuss invariant properties, as well as to reason with and about them. We also saw that teachers’ backgrounds and past experiences can play an important role in their descriptions of invariant properties. Possible future research directions and implications to teacher education are discussed.

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Data availability

The data that support the findings of this study are available on request from the corresponding author [G.G.N]. The data are not publicly available because the data contain information that could compromise the privacy of research participants.

Code availability

Not applicable.

Notes

  1. To read more about the Geometer’s Sketchpad® and the Sketchpad Explorer, go to https://www.dynamicgeometry.com/General_Resources/Sketchpad_Explorer_for_iPad.html.

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Acknowledgements

The research reported here was supported in part by the Dean of the College of Arts and Sciences, at the University of Massachusetts Dartmouth (UMass D), through a Graduate Thesis Support Grant to the authors. The opinions expressed are those of the authors and do not represent views of UMass D.

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Partial support was received from the Dean of the College of Arts and Sciences, at the University of Massachusetts Dartmouth, through a Graduate Thesis Support Grant to the authors.

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Authors and Affiliations

  1. University of Massachusetts Dartmouth, 285 Old Westport Road, Liberal Arts 398F, Dartmouth, MA, 02747, USA

    Gili Gal Nagar & Chandra Hawley Orrill

  2. Southern Connecticut State University, 501 Crescent Street, New Haven, CT, 06515, USA

    Stephen Hegedus

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  1. Gili Gal Nagar
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Correspondence to Gili Gal Nagar.

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Nagar, G.G., Hegedus, S. & Orrill, C.H. High school mathematics teachers’ discernment of invariant properties in a dynamic geometry environment. Educ Stud Math 111, 127–145 (2022). https://doi.org/10.1007/s10649-022-10144-6

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