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.
Thursday, March 28, 2019
Parabola Paradise
There are many things you can create using any line in space and any point not on the line. One of which is a parabola! In fact, with every line and every point not on that line, there exists a unique parabola. The parabola is created by connecting the accumulation of the infinite number of points that lie exactly equidistant to the line (the shortest distance to this line) and the given point. Parabolas of varying narrownesses and widenesses have been created and analyzed as a result of this construction. It was fascinating to me to construct a parabola in this way in Geometer's Sketchpad alongside some of my classmates who are all studying how to best use technology in the classroom as future educators.
Once again, you will have to download Geometer's Sketchpad in order to view the attached exploration.
Once again, you will have to download Geometer's Sketchpad in order to view the attached exploration.
Wednesday, March 27, 2019
Creating Quadrilaterals From Quadrilaterals
Summary:
In preparation for a discussion in class on Wayne Nirode’s article, “Creating Quadrilaterals from Quadrilaterals,” (the link takes you to a place where you can choose to purchase the full article) I read it and took notes that provided a lot of insights into how to best create tasks that engage my students and enlarge their minds, no matter the level of intellect. This quadrilaterals task began with students creating new convex quadrilaterals from the given convex quadrilaterals by connecting the four midpoints of each side in a successive order, for example. Each student was asked to create his or her own 10 rules for creating new quadrilaterals, and by the time the class gathered together all of their rules, there were up to 150 unique ideas! With all of these rules, groups of students were assigned a few to prove on their own, analyze each of them, and create posters presenting the rule and its proof. Each proof analyzed which kind of quadrilateral could be made using the given rule based on the starting quadrilateral. As I read about this task and the successes that came from it, I learned that I could create tasks with both a low floor and a high ceiling (LFHC). LFHC is used to describe educational tasks that are easy to begin and think about at first, but that the same task could encourage deeper thinking within students who finish quickly, or quickly learn the fundamental mathematical concepts for the task. Creating Quadrilaterals from Quadrilaterals is the perfect example of a LFHC task.
The professor and students studying to be educators in the future in my Teaching With Technology class at Brigham Young University (BYU) extensively discussed ideas that stemmed from this task. While we talked a little bit about the benefits of the quadrilaterals task, we branched off for the majority of the discussion and shared ideas about the best ways we could implement geometric proofs in our future classrooms. The task given provided multiple opportunities for students to grapple with different quadrilaterals and trying to prove them. If they got stuck, the task gave an opportunity for students to move onto a different rule to try to prove together. If proofs are wrong, students can discuss all the ways instead of just the right ways to complete the proofs. This would enlarge their minds as they consider why certain reasoning is wrong. This will surely solidify their understanding of why certain proofs are correct as well. In addition, my classmates and I all reminded each other about some critical geometric definitions, such as orthocenters, centroids, and incenters, and how they all relate to exploring relationships with triangles. Someone noted that if there is this much variety (and more) in exploration of characteristics of triangles in geometry, how many more characteristics could be explored in quadrilateral characteristics!
Critique:
In our class discussion, it was mentioned that students will distract themselves with anything, no matter what task you give them, and that the only thing you can do to help engage the students is to give them an especially engaging task. While I do believe this is helpful (and vital) in keeping students’ attention, I also believe my responsibility to help them stay on task only goes so far. I want to instill within each of students a personal responsibility and accountabliltiy for their learning. However, I also believe that it is my job as their teacher to create the best environment possible to facilitate meaningful learning experiences. I can do this with what I put on my walls, my established rules in the classroom, and a trusting teacher-student relationship with each pupil.
While I love the part of the task when students are able to get together in a group and prove different rules for creating new quadrilaterals because they can bounce ideas off of one another, I think this task would be far too long, and students might start to not like it after about 2 class periods. Making posters fosters creativity and engages the right side of the brain in a mathematics classroom—which does not happen very often! However, I believe making posters would take up too much valuable time in the classroom.
Connections:
The article provided an excellent example of the kinds of tasks I want to use in my future classroom. This task is similar to a practice we have discussed as a class in previous discussions that I would love to implement—that is, I would like to use the textbook as a resource in creating meaningful tasks for students as I search the example problems in the back or at the end of the section, and begin my lesson with a question towards the end. These questions are written for an audience who has already learned the material and is simply thinking about the concepts deeper for further enlightenment. If I choose to begin my lessons with these types of questions, students can grapple with it and come to their own conclusions. They can seek to learn the processes and procedures on their own with a facilitated discussion. This task seems to be something Nirode took from the back of a textbook to begin with, and ran with it, hoping students would learn powerful facts and concepts about quadrilaterals through their own exploration of it rather than a list of formulas and factual characteristics of quadrilaterals.
Something profound that my professor suggested was to never force students to prove conjectures using the traditional two-column format, where the first column involves the line of logical statements from beginning to end and the second column includes the reasons why the logical statement can be said. In his extensive experience teaching geometry to high school students, he found that they would often write down the “given” and the conclusion but not understand what to write in between. The point of the proof is to encourage creating an argument as to why a certain conjecture is correct (or incorrect, as discussed previously), and not in arguing through a specified two-column format that often gets in the way of the desired purpose of geometric proofs. This discussion changed how I plan on teaching geometric proofs in the future. It is refreshing to remember the true purpose of a proof and how my teaching strategies and tasks can illustrate that.
In preparation for a discussion in class on Wayne Nirode’s article, “Creating Quadrilaterals from Quadrilaterals,” (the link takes you to a place where you can choose to purchase the full article) I read it and took notes that provided a lot of insights into how to best create tasks that engage my students and enlarge their minds, no matter the level of intellect. This quadrilaterals task began with students creating new convex quadrilaterals from the given convex quadrilaterals by connecting the four midpoints of each side in a successive order, for example. Each student was asked to create his or her own 10 rules for creating new quadrilaterals, and by the time the class gathered together all of their rules, there were up to 150 unique ideas! With all of these rules, groups of students were assigned a few to prove on their own, analyze each of them, and create posters presenting the rule and its proof. Each proof analyzed which kind of quadrilateral could be made using the given rule based on the starting quadrilateral. As I read about this task and the successes that came from it, I learned that I could create tasks with both a low floor and a high ceiling (LFHC). LFHC is used to describe educational tasks that are easy to begin and think about at first, but that the same task could encourage deeper thinking within students who finish quickly, or quickly learn the fundamental mathematical concepts for the task. Creating Quadrilaterals from Quadrilaterals is the perfect example of a LFHC task.
The professor and students studying to be educators in the future in my Teaching With Technology class at Brigham Young University (BYU) extensively discussed ideas that stemmed from this task. While we talked a little bit about the benefits of the quadrilaterals task, we branched off for the majority of the discussion and shared ideas about the best ways we could implement geometric proofs in our future classrooms. The task given provided multiple opportunities for students to grapple with different quadrilaterals and trying to prove them. If they got stuck, the task gave an opportunity for students to move onto a different rule to try to prove together. If proofs are wrong, students can discuss all the ways instead of just the right ways to complete the proofs. This would enlarge their minds as they consider why certain reasoning is wrong. This will surely solidify their understanding of why certain proofs are correct as well. In addition, my classmates and I all reminded each other about some critical geometric definitions, such as orthocenters, centroids, and incenters, and how they all relate to exploring relationships with triangles. Someone noted that if there is this much variety (and more) in exploration of characteristics of triangles in geometry, how many more characteristics could be explored in quadrilateral characteristics!
Critique:
In our class discussion, it was mentioned that students will distract themselves with anything, no matter what task you give them, and that the only thing you can do to help engage the students is to give them an especially engaging task. While I do believe this is helpful (and vital) in keeping students’ attention, I also believe my responsibility to help them stay on task only goes so far. I want to instill within each of students a personal responsibility and accountabliltiy for their learning. However, I also believe that it is my job as their teacher to create the best environment possible to facilitate meaningful learning experiences. I can do this with what I put on my walls, my established rules in the classroom, and a trusting teacher-student relationship with each pupil.
While I love the part of the task when students are able to get together in a group and prove different rules for creating new quadrilaterals because they can bounce ideas off of one another, I think this task would be far too long, and students might start to not like it after about 2 class periods. Making posters fosters creativity and engages the right side of the brain in a mathematics classroom—which does not happen very often! However, I believe making posters would take up too much valuable time in the classroom.
Connections:
The article provided an excellent example of the kinds of tasks I want to use in my future classroom. This task is similar to a practice we have discussed as a class in previous discussions that I would love to implement—that is, I would like to use the textbook as a resource in creating meaningful tasks for students as I search the example problems in the back or at the end of the section, and begin my lesson with a question towards the end. These questions are written for an audience who has already learned the material and is simply thinking about the concepts deeper for further enlightenment. If I choose to begin my lessons with these types of questions, students can grapple with it and come to their own conclusions. They can seek to learn the processes and procedures on their own with a facilitated discussion. This task seems to be something Nirode took from the back of a textbook to begin with, and ran with it, hoping students would learn powerful facts and concepts about quadrilaterals through their own exploration of it rather than a list of formulas and factual characteristics of quadrilaterals.
Something profound that my professor suggested was to never force students to prove conjectures using the traditional two-column format, where the first column involves the line of logical statements from beginning to end and the second column includes the reasons why the logical statement can be said. In his extensive experience teaching geometry to high school students, he found that they would often write down the “given” and the conclusion but not understand what to write in between. The point of the proof is to encourage creating an argument as to why a certain conjecture is correct (or incorrect, as discussed previously), and not in arguing through a specified two-column format that often gets in the way of the desired purpose of geometric proofs. This discussion changed how I plan on teaching geometric proofs in the future. It is refreshing to remember the true purpose of a proof and how my teaching strategies and tasks can illustrate that.
Monday, March 11, 2019
Geometer’s Sketchpad Toolkit
There’s a handy program called Geometer’s Sketchpad that is especially important to future math educators in constructing geometric shapes quickly and easily using digital versions of a compass and a straightedge. For example, one can quickly create a regular hexagon given the length of one of its sides, or a right isosceles triangle given one of the legs. While staying in Euclidean geometry, I was able to construct 10 tools that are saved onto one Sketchpad document.
Unfortunately, you will need to download Geometer’s Sketchpad to access this “toolbox” of mine. It is fairly easy and does not take up a lot of storage space. You will need a desktop or laptop computer, as a downloadable version of Geometer’s Sketchpad for tablets and mobile devices either does not exist or is not yet available to the public.
Happy geometering!
Unfortunately, you will need to download Geometer’s Sketchpad to access this “toolbox” of mine. It is fairly easy and does not take up a lot of storage space. You will need a desktop or laptop computer, as a downloadable version of Geometer’s Sketchpad for tablets and mobile devices either does not exist or is not yet available to the public.
Happy geometering!
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