Understanding Geometric Transformations

Summary:

In preparation for a discussion in class on Karen F. Hollebrands’ article “High School Students’ Intuitive Understandings of Geometric Transformations,” I read it and took notes that provided a lot of insights into the research that was conducted and how it relates to my future mathematics classroom. High school students who had taken algebra 1 in eighth grade were chosen in order to analyze how much knowledge of geometric transformations they were able to remember. The geometric transformations in question included reflections, rotations, and translations. Many conclusions were analyzed and discussed, such as the fact that students seemed to think about reflections as flipping an image over a certain line instead of that line being the perpendicular bisector of all the lines connecting each point of the image on each side of the line.

The professor and students studying to be educators in the future in my Teaching With Technology class at Brigham Young University (BYU) extensively discussed the ideas presented in Hollebrands’ article. We discussed the implications of reflecting an entire plane of points rather than just the preimage. This idea is not typically seen when analyzing how students perceive reflections, so this was an interesting conjecture—something I have never thought about before! We talked about things we learned from the article, like reflecting points across a line of reflection simply means the points are equidistant from the line. The same thing happens when reflecting lines or entire images—each part of the line or image is exactly the same distance from the line of symmetry as its preimage.

Critique:

It was mentioned in the article that the transformations that the students struggled the most with were translations. This was due to the complete inability to appropriately use the given vector describing the translation required. Because of this, I wholeheartedly disagree with the conclusion the author made from the results of this study—that students understood translations the least simply because they could not use vectors exactly how they were intended to be used for the transformations. I believe this was not an accurate conclusion that could be made from the information gained from the study.

In addition, it was stated that students are typically more successful when the transformation requires a reflection over horizontal or vertical lines. This reminds me that rotations are usually easier to visualize (from my own experience) when it requires 90, 180, 270, or 360 degree-rotations instead of any other rotations, such as 45-degrees or 135-degrees. This was an intuitive conclusion made from the study’s results, and I agree with it.

Connections:

It was stated that students who heavily relied on the visual aspect of the preimage instead of the specific properties that define certain transformations were more likely to have transformed the preimage incorrectly. Knowing this fact, I plan on teaching the fundamental properties of each transformation first in the unit so students understand the underlying mathematics of each one instead of an artsy new configuration of a picture. Most of my experience with transformations growing up in school involved flipping or turning an image on its side. I never learned the actual properties of transformations—such as each point being equidistant from the line of reflection as its preimage. Knowing this fact would not only help me visualize the new image better (even if the line of reflection was not vertical or horizontal), but it would help me to check my answer mathematically to ensure accuracy. Overall, I learned a great amount about transformations as I analyzed the study done with these brave eighth graders.