I read the article High School Students’ Intuitive Understandings of Geometric Transformations. Here are my notes.
Page 1:
This article focuses on analyzing students’ work on tasks related to the translations, reflections, and rotations. One type of task asked students to draw the image of a polygon under a reflection, rotation, or translation.
Page 2:
When students were asked to define a reflection, the most common response was, “a reflection is like a mirror.”
Students encountered difficulties in accurately placing image quadrilaterals.
All students seemed to think about reflection as flipping, and they explained that they tried to draw the image congruent to the preimage; however, they did not attend to relationships between corresponding preimage and image points and the line of reflection.
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rotation-> most described it as turning an object
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Although students seemed to associate rotation with turning, they used the center of rotation in different ways. One student appeared to use point B as the center of rotation and turned the triangle until point A coincided with point O, as shown in figure 5a. Another student, who explained that she turned triangle ABC 45 degrees, appeared to be using point B as the center of rotation and then translated the triangle so that point B coincided with point O, as shown in figure 5b. Most students who used the center of rotation appropriately did not attend to the property that corresponding preimage and image points are equidistant from the center of rotation.
These errors suggest that teachers should provide opportunities for students to attend to relationships among the center, preimage, and corresponding image points.
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Students had the hardest times with translations
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These results differ from other researchers’ findings about students’ understandings of translations. On tasks for young children that involved physical objects, both Schultz and Austin (1983) and Moyer (1978) reported that students were more successful in performing translation tasks than tasks involving reflections or rotations. However, students were not given a specific translating vector to use to perform the translation in either of those studies. The presence of the vector may have confused students.
All of Page 7
Page 8:
Students were asked to explore the effects of changing the positions of the points or lines to experience how changing a preimage point affects the image point and how the images change if the line of reflection is moved. This activity helped students understand the role of the line of reflection. To focus students on relationships between the line of reflection and the preimage and image points, students were asked to create LL`', MM`', and NN`' and the points where these segments intersected the line of reflection, points P, Q, and R, respectively.
For rotations, I chose to use a compass to focus students’ attention on the importance of the center of rotation and the fact that corresponding preimage and image points are equidistant from the center of rotation. I used The Geometer’s Sketchpad again for translations to focus students on the importance of the vector and that it was parallel to and equal in measure to segments joining corresponding preimage and image points.