Second day in a row here for the blog, today we will be discussing an article found in the NCTM publication from March 2004. This is quite a few years ago, in fact, this article is even older than some current secondary students! Which is kind of insane, but hopefully the information given in the article is still valid and helpful. The article can be found at the following link. Thankfully, this link doesn't require you to have any special access, anyone should be able to follow this! Good news!
Summary:
This article, written by Karen Hollebrands, describes a study of her high school student's understandings of the rigid motion transformations (translations, reflections, and rotations). The article then goes through each of these transformations and students understanding of these transformations. In short, students understood that reflections changed the orientation of the shapes but didn't completely understand how the line of reflection determined a unique image from a preimage. Students seemed to understand how a rotation works, but missed that a shape can be rotated around a single point and how that works. In her study, students understood that shapes can be translated but were unsure of how to define a unique image from the preimage. Overall, it seemed as though students understood that these three transformation maintained size, shapes, and angles but struggled to determine exactly how they were uniquely defined from the information given.
As I discussed this with a few of my peers, we found a few problems with this study. First of all, we found that while Karen Hollenbrands claimed that she was testing her students' understanding of the transformations she may not have actually been testing that, but testing their knowledge of notation and things that she had yet to teach. While it is a good idea to try to understand where students are in their understanding before we teach, we have to be sure that we are accurately representing their true understandings. One of the major issues we brought up was that when asking students to translate a figure they were given a translation vector, something that students had never seen before. It seemed that she was actually testing her students' knowledge of vectors, not of translations.
However, our discussion wasn't only about the faults that we found with the article. We discussed how the article suggested that students be given access to Geometer's Sketch Pad in order to understand transformations. This tool allows students to try a huge variety of different transformations in a short amount of time, allowing them to experiment and understand what the relationship between an image and preimage is for any given translation. It also helps students understand what is necessary for any given translation. We then spent some time discussing what is necessary to define a translation and what our students might believe about these transformations.
Critique:
The class discussion of this article fairly accurately sums up my thoughts on the article. I feel that Karen Hollenbrands didn't accurately portray her student's thinking with what she tested. It seems that the ideas and concepts of testing student understanding could be extremely useful but that the methodology of testing understanding would need to be improved in order to accurately represent student understanding. I do feel like Geometer's Sketch Pad is an extremely useful tool, not just for what has been discussed in previous blog posts, but also for transformational geometry. Students can use the dynamic capabilities of the software to represent a variety of transformations and really understand how a single change in the "givens" of a transformation can change the result. The article and discussion following the reading of the article really drew that out and helped me to understand the many uses that a dynamic software can have for helping student understanding.
Connections:
In my mind, students understanding that these 3 transformations are rigid motion transformations is an extremely important piece of student knowledge and the simple part of teaching about lines of reflection, centers of rotation, and how to represent translations is not as important initially. If students understand these, I feel that in my practice as a teacher I will be able to solidify and bring together all these ideas into what is mathematical practice.
Something that is worrisome to me is that there are many extremely useful tools that I can use in my future classroom, but I need to figure out which ones to use and figure out a way to have them all in my classroom. Geometer's Sketch Pad can be an expensive tool, and if I need to have this and a variety of other programs on a classroom set of computers that cost could add up quickly. I don't know how to petition administrators to purchase a variety of softwares or other technology to further learning, and I don't know how to learn exactly how to learn that either.
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