In recent years, a transformation perspective has come to the fore in K-12 standards (National Governors Association Center for Best Practices, Council of Chief State School Officers, 2010; NCTM, 2018) leading many college instructors to transition their geometry for teachers courses to include transformation geometry. Unfortunately, at any level, there has been “limited research explicitly on the topics of congruency and similarity, and little on transformation geometry” (Jones & Tzekaki, 2016, p. 139).

In order to support instructors working to incorporate a transformation context into their course, Dr. Yvonne Lai (University of Nebraska-Lincoln) and I have worked to more carefully understand student thinking on transformation congruence proofs (St. Goar & Lai, 2019, 2020). Much of our work focuses on how students understand and use the definition of congruence, which states that two figures are considered congruent if, and only if, there exists a sequence of transformations (translations, reflections, and rotations) that maps one object to the other.

In the course of our research so far, we have analyzed student work on transformation congruence proofs from homework and exams in two geometry for teachers courses; one taught at UNL and the other at Merrimack College. We have analyzed student work on many different problems, and student work on the Boomerang Problem and the Line-Point Problem (Figure 1) proved to be particularly enlightening for this research.

Our analysis uncovered possible key developmental understandings (Simon, 2006) as students work to employ the definition of congruence in proof. As a brief summary, a key developmental understanding (KDU) affords the learner a different way of thinking about mathematical relationships, and learners acquire KDUs through reflection and multiple experiences – not by direct explanation. As a result, special care may be needed to reinforce KDUs repeatedly throughout a term and in many different contexts. In a transformation congruence proof context, we identified the following potential KDU’s:

1. Understanding that applying the definition of congruence to prove congruence of two figures means establishing a sequence of rigid motions mapping one entire figure to the other entire figure.
2. Understanding that using a sequence of transformations to prove that two figures are congruent means justifying deductively that the image of one figure under the sequence of transformations is exactly the other figure.

To clarify, a student who doesn’t show evidence of the first potential KDU above may still know that rigid motions are involved in a transformation congruence proof. However, they may not realize that a figure must remain intact and unaltered throughout the transformation process. For example, when looking at student work on the Boomerang Problem, we found that some students would apply different rigid motions to each triangle. Some of these students might then conclude that because ∆ABC≅∆DEF and ∆ABO≅∆DEP, that the union of these triangles is also congruent.   

Understanding the second potential KDU involves realizing that successfully identifying a candidate sequence of transformations between two figures isn’t sufficient to demonstrate congruence between the two figures. For example, on the line-point problem, some students correctly constructed sequences of rigid motions from one entire figure (i.e. the union of the line and the point) to the other entire figure, yet the students stopped short of making an argument explaining why the rigid motions caused the two figures to be superimposed. In some of these cases, we suspect students are viewing these figures as already superimposed and thus may not grasp why this justification is necessary.

This research has helped us to illuminate the perspectives of the students in our geometry for teachers courses to provide better feedback to students during group work and, when correcting their work,  to become aware of concepts that need to be repeatedly reinforced in lesson plans and assignments. We hope readers will be able to do the same and to find new ways to apply these perspectives to their courses. 

Acknowledgements

This work is partially supported by NSF DUE- 1937512. 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. 

References

Jones, K., & Tzekaki, M. (2016). Research on the teaching and learning of geometry. In A. Gutiérrez, G. Leder & P. Boero (Eds.), The Second Handbook of Research on the Psychology of Mathematics Education: The Journey Continues (pp. 109-149). Rotterdam: Sense.

National Council of Teachers of Mathematics. (2018). Catalyzing change in high school mathematics: Initiating critical conversations. Reston, VA: Author.

National Governors Association Center for Best Practices, Council of Chief State School Officers. (2010). Common Core State Standards for mathematics. Washington, DC: Author.

Simon, M. A. (2006). Key developmental understandings in mathematics: A direction for investigating and establishing learning goals. Mathematical Thinking and Learning, 8(4), 359-371.

St. Goar, J., & Lai, Y. (2019). Prospective High School Teachers Understanding and Application of the Connection Between Congruence and Transformation in Congruence Proofs. Weinberg, A., Moore-Russo, D., Soto, H., & Wawro, M. (Eds.), Proceedings of the 22nd Annual Conference on Research on Undergraduate Mathematics Education, (pp. 247-254). Oklahoma City, OK.

St. Goar, J. & Lai, Y. (2020). Defining Key Developmental Understandings in Congruence Proofs from a Transformation Approach. Karunakaran, S., Reed, Z., & Higgins, A. (Eds.), Proceedings of the 23^rd Annual Conference on Research in Undergraduate Mathematics Education, (pp. 880-885). Boston, MA.