Creating Quadrilaterals from Existing Quadrilaterals

So here goes. The next few days are going to see more posts here than there have been for a few weeks. Life has been crazy and full of homework, and I've finally found the time to put together a few more posts.

Today I'd like to discuss an article titled "Creating Quadrilaterals from Quadrilaterals" that can be found here. It's an article written in 2016, but contains some interesting ideas about creatively implementing the Common Core curriculum using dynamic software.

Summary:

In this article, Wayne Nirode presents an interesting lesson that he designed around helping students explore a variety of student made conjectures about quadrilaterals. In essence, over the course of a few days, students created their own ways to make new quadrilaterals from existing quadrilaterals, observed how these new quadrilaterals were related to the old using dynamic geometric software, and finally attempting to prove these conjectures in a rigorous/semi-rigorous fashion. This created a very useful environment where students needed to use and develop vocabulary and proof skills in a creative and fun way. This activity exceeded the demands of the Common Core curriculum, but succeeded in helping students stay engaged in the mathematics and truly exploring.

As I discussed this article with a few of my peers we found some interesting points that Wayne Nirode didn't specifically pull out of the activity. Something that we found interesting was that this could be a fairly vulnerable activity as a teacher. In this activity, students were asked to create their own conjectures and really engage in mathematical research. The teacher provides scaffolding for the initial exploration with the three examples of how to create new quadrilaterals from an existing quadrilateral and asked the students to explore from there. In response, the students in the article created 150 unique constructions, allowing for a huge pool of mathematical conjectures and actual mathematical practice. When students come across these conjectures, they are extremely likely to struggle with the mathematics and may in fact be unable to prove using the tools they have. As teachers we have the duty to allow these students to struggle with the mathematics enough to promote understanding but not enough that they become discouraged. This can take a variety of different forms, and in my view is possibly the most difficult part about being a math teacher.

Critique:

One problem that I have with this activity is the amount of time that this activity requires. Students were given a few days in class to create their constructions, a few days to create conjectures about these, and a few more days to create and present their proofs. All in all, this seems as though it would take at least a week of daily classes in order to finish this activity. While this seems like quite a bit of time, students are extremely likely to come out of this activity with having an understanding of special quadrilaterals (squares, trapezoids, kites, etc) and have some practice with creating a variety of proofs and struggling with mathematics. However, it seems that if a teacher isn't careful with managing time and helping students stay on task a lot of time could be wasted.

As a group, we also brought to light an interesting dilemma about engaging students in mathematical discourse. Can we really engage all our students in true mathematical struggle? What would happen if one student becomes distracted and begins to distract others from the mathematics? How do we balance the need for exploration and the need for discipline in our mathematics students? These questions are very interesting and fall into the realm of classroom management. I don't have all/any answers for these, I'll leave these answers for another post/author.

Connections:

Something that really stuck out to me from this article is the usefulness and necessary aspect of dynamic geometric software for this exploration. It would be almost impossible to do a similar exploration without software, students would need to create hundreds of examples by hand, something that could be done in a few minutes using a computer. One of my biggest worries about having technology in my classroom is that students could easily be distracted and get lost in their Facebook feed rather being lost into the mathematics. This article and many others like it have helped me to understand the extreme benefit that technology can bring into a mathematics classroom, largely outweighing the potential disadvantages of technology.

I wonder if there are great advantages to having phones in class. I've largely seen phones as distractions in math classes, but at the same time I have a graphing calculator on my phone. That is extremely useful for me, but as a teacher I'm initially reluctant to give students free access to their phones in my classroom. I guess that's going to be a problem that I will need to address with my school and fellow teachers on what my expectations should be in my classroom.