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.