Summary of Article:
I read the article High School Students’ Intuitive Understandings of Geometric Transformations. This article explains how a teacher, Karen Hollebrands, chooses to teach her 10th grade geometry students about geometric transformations. She started by finding out what they already knew by conducting interviews of six of her 17 students before, during and after the seven week unit. Although she already had lesson plans, she was able to use the information she gained form the interviews to refine her lessons. Then she describes her students during the seven week unit. Her students were most familiar with reflections. Most students knew that the image had to be flipped, but didn't take into consideration the other properties about the flip, including the perpendicular bisectors. By asking her students what they were thinking Karen was able to understand why students were making mistakes. When the students moved onto rotations, although some knew what was meant by a center of rotation, not all did. And although some thought they knew what it meant, they had a hard time using it. And those students that did do it right didn't take into consideration the other properties about the rotation, including the fact that corresponding pre image, and image points are equidistant from the center of rotation. Like with the reflection, having students explain what they did helped the teacher see where they were making mistakes. The third transformation was the translation. This was the most difficult transformation for the students. The students hadn't dealt with vectors, and were unsure of how to use them in the translations. The students could understand that the figure needed to be slid, but they didn't think of the vector as having direction and magnitude. The article mentions that GSP can be useful when discussing vectors because it can show how things move.
Research had shown that the translations would be the easiest transformation, but through interview Karen saw that they ended up being the most difficult for the class. Her interviews also showed her that although students could conceptualize the idea of the transformations, they didn't take into consideration the parameters that go along with the transformations. Karen used the GSP to initially teach students about the different transformations "because [she] thought that doing so would help focus students’ attention on the line of reflection and on relations between the line of reflection and corresponding preimage and image points." Karen concluded her article with a great summary: "The use of interviewing provided valuable information about my students’ understandings of geometric transformations. Analyzing the interviews enabled me to make instructional decisions that were informed by my knowledge of my students’ understandings so that I could build on their current conceptions. Although interviewing every student may not always be feasible, findings from research about students’ understandings may provide a starting point for making instructional decisions."
Summary of Class Discussion:
We started by talking about the benefit of knowing in advance the difficulties that students may experience. Karen knew the difficulties in advance through her interviews. We compared this teaching method to that of the one in the TI nspire case study. That teacher gave his students a pre test, but then just taught the students what they missed, where as this teacher, Karen, taught the students everything, and just made sure to highlight the things that students didn't understand. Knowing difficulties in advanced can make a difference, but it depends on what you do with that knowledge, it makes a different of how valuable the mathematics actually is for the students. We then talked about properties of transformations. For example, translations using vectors. When I was in high school I was taught to just slide the figure, I wasn't taught about the vectors that were moving in the translation, but this teacher makes sure to point them out. The location of a vector doesn't change its direction or magnitude, so the transformations shouldn't either. Another property is that transformations transform the entire plane, not just the figure, but students don't always know that. Also, a transformation maps everything from the entire plane back onto the entire plane, so the entire plane has to be in the domain and range. A third property of transformations are their parameters. Dr. Leatham called them part of a function, not parameters.
To bring up a different subject, Joe asked why we work so hard to make things familiar for the students. We then talked about the paradox of learning: in order to learn something you have to build on something else, so the instant that you make an association with something you know about, it's easy to make a one to one correlation, but that relationship could be wrong. A good way to think of how we learn is that we start off over generalizing things, and then we eventually narrow our conceptions, and make distinctions. Learning is the process of refining generalizations.
Because this article mentioned a lot of research that has been done regarding geometry, someone in the class asked how we as teachers can access this research. We can find articles like the one we just read in mathematical journals, workshops, nctm.org, and other places. A dilema is that researchers are encouraged to share research with others who are doing research, but not really the people who could use the research, so it's not as accessible for teachers as it is for researchers.
We concluded by talking about how we can apply this article to our own teaching. The question was asked, what's one piece about students misconceptions that we got from this article? A few things that our class mentioned were making sure to define things well, the tool determines a lot of the work that will be done (for example it is easier to show a center of rotation on GSP than on paper). We need to remember that creating connections is just as important as learning new material. Because of the paradox of learning, students have to build on what they already know, so it's good to find out what they know and build on that.
Critique:
I wish that we had talked more about the research that Karen, the teacher, talked about. She compared some of her findings to research that had previously been done, but in our class discussion we didn't talk much abtou how her results compared to the research. With regard to the article, how would the lessons have been different without GSP? GSP helps students see where the point of rotation is for a figure, but it doesn't really allow them to make mistakes the way paper and pencil would. I wonder if it would have been beneficial for students to try things on paper and pencil, then go to GSP. For example, students rotate or reflect a figure on paper, then go to GSP where all they have to do is select the point of rotation, or the mirror, and transform the figure. They might be able to see if and how their paper and pencil rendition is different than GSP.
Connections:
What I really got out of this article is that we can't just assume that our class will be like the norm. Research had told this teacher that translations would be the easiest transformation for her class, but they ended up being the most difficult. Also, with relation to that, we can't assume that one class will be similar to the other. I was glad to see a pre assessment (the interview) that was used to help the teacher know what to teach, but she didn't limit her teaching just to that. She made sure to cover all the material, but she knew where there might be problems, so she was aware a head of time that she would probably need more time, or specific tasks to flush out those mistakes. I think that a pre assessment of some sort would be a really be a good idea when used correctly. I wonder if Karen had done a pretest, like the TI nspire case study, if she would have been able to get as much information as she did from interviews where she could actually ask the students what they were thinking. I think the interview would give more information, but its more difficult to interview everyone, where are its easy to pre test everyone. Those will be things that I have to think about when I am a teacher.