Parabolas for days

Here is an interesting exploration about parabolas. This involves a geometric and analytic dynamic representation of a parabola, including analysis of the directrix, focus, line of reflection, and vertex. Another very interesting piece of this exploration is how the vertex changes as a, b, and c vary from our general equation y = ax^2 + bx + c.

This is a gsp file, so in order to open this you must have Geometer Sketch Pad. Sorry for everyone out there who doesn't.

Here is the file. Have fun exploring!

Secondary Student's Understanding of Geometric Transformations

Second day in a row here for the blog, today we will be discussing an article found in the NCTM publication from March 2004. This is quite a few years ago, in fact, this article is even older than some current secondary students! Which is kind of insane, but hopefully the information given in the article is still valid and helpful. The article can be found at the following link. Thankfully, this link doesn't require you to have any special access, anyone should be able to follow this! Good news!

Summary:
This article, written by Karen Hollebrands, describes a study of her high school student's understandings of the rigid motion transformations (translations, reflections, and rotations). The article then goes through each of these transformations and students understanding of these transformations. In short, students understood that reflections changed the orientation of the shapes but didn't completely understand how the line of reflection determined a unique image from a preimage. Students seemed to understand how a rotation works, but missed that a shape can be rotated around a single point and how that works. In her study, students understood that shapes can be translated but were unsure of how to define a unique image from the preimage. Overall, it seemed as though students understood that these three transformation maintained size, shapes, and angles but struggled to determine exactly how they were uniquely defined from the information given.

As I discussed this with a few of my peers, we found a few problems with this study. First of all, we found that while Karen Hollenbrands claimed that she was testing her students' understanding of the transformations she may not have actually been testing that, but testing their knowledge of notation and things that she had yet to teach. While it is a good idea to try to understand where students are in their understanding before we teach, we have to be sure that we are accurately representing their true understandings. One of the major issues we brought up was that when asking students to translate a figure they were given a translation vector, something that students had never seen before. It seemed that she was actually testing her students' knowledge of vectors, not of translations.
However, our discussion wasn't only about the faults that we found with the article. We discussed how the article suggested that students be given access to Geometer's Sketch Pad in order to understand transformations. This tool allows students to try a huge variety of different transformations in a short amount of time, allowing them to experiment and understand what the relationship between an image and preimage is for any given translation. It also helps students understand what is necessary for any given translation. We then spent some time discussing what is necessary to define a translation and what our students might believe about these transformations.

Critique:
The class discussion of this article fairly accurately sums up my thoughts on the article. I feel that Karen Hollenbrands didn't accurately portray her student's thinking with what she tested. It seems that the ideas and concepts of testing student understanding could be extremely useful but that the methodology of testing understanding would need to be improved in order to accurately represent student understanding. I do feel like Geometer's Sketch Pad is an extremely useful tool, not just for what has been discussed in previous blog posts, but also for transformational geometry. Students can use the dynamic capabilities of the software to represent a variety of transformations and really understand how a single change in the "givens" of a transformation can change the result. The article and discussion following the reading of the article really drew that out and helped me to understand the many uses that a dynamic software can have for helping student understanding.

Connections:
In my mind, students understanding that these 3 transformations are rigid motion transformations is an extremely important piece of student knowledge and the simple part of teaching about lines of reflection, centers of rotation, and how to represent translations is not as important initially. If students understand these, I feel that in my practice as a teacher I will be able to solidify and bring together all these ideas into what is mathematical practice.

Something that is worrisome to me is that there are many extremely useful tools that I can use in my future classroom, but I need to figure out which ones to use and figure out a way to have them all in my classroom. Geometer's Sketch Pad can be an expensive tool, and if I need to have this and a variety of other programs on a classroom set of computers that cost could add up quickly. I don't know how to petition administrators to purchase a variety of softwares or other technology to further learning, and I don't know how to learn exactly how to learn that either.

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.

Geometer's Sketch Pad

It's been a while since my last post, and this is a pretty small post. Sorry to the one dedicated reader of my blog. There are going to be more posts in the near future.

I've recently been playing with Geometer's Sketch Pad. If you haven't ever played with it, I highly recommend it. It's been a lot of fun, even as I've run into some limitations of the software. But some of the most fun things that I've done is to create a bunch of constructions. Software like this allows the user to go through thousands of creations and dynamically view a construction in many, many ways. And if you make a single mistake (like I do often) you don't have to completely erase your entire drawing. Instead, just hit Ctrl Z and go back to having fun.

Anyway, here are a few of my most recent constructions in GSP. They are all created as tools, and are fairly cool. If you want to see how they were made, you can view the script view and walk through the entire construction.

Here is a link to a copy of my gsp file (you'll need GSP in order to view these, sadly).

https://drive.google.com/open?id=1r4mseSTA-rgLa3ez4TTFAXd6axeayZNG