Conference Connections

This past week was General Conference. This is a twice yearly event where we as members of the Church of Jesus Christ of Latter-Day Saints have the opportunity to listen to our prophet and apostles (and other leaders) speak about spiritual topics. This often includes teaching about Christ's life, how we can become more like Him, invitations to repent, and many more spiritual ideas.

Within one of my classes, we decided to all watch, read, listen to, or participate in General Conference in some way and specifically focus on how the teachings of the Gospel of Jesus Christ apply to our future math classrooms. This was a very interesting focus, and I feel like the principles that we pulled are applicable to all math teachers, not just those who may be members of our faith. 

There is a problem, though. Conference was a total of 10 hours, and full of connections to education and technology. I'm going to spend the next few months going over all the talks again and again, and picking a single talk to focus on is quite difficult to do. This is part of the reason why this reflection is so late, I had a difficult time narrowing my focus to something that would fit in a single blog post.

I chose to focus on one single talk, Elder Bednar's talk Preparing to Obtain Every Needful Thing. Elder David A Bednar is actually an educator by profession, and acted as a professor, an associate dean, and as president of Rick's College. It's not surprising, then that there are many, many connections to education to be found in his talks. This talk is no exception. In this post, I will go through some of my favorite quotes that I found and relate them directly to our math classrooms.

"We should not expect the Church as an organization to teach or tell us everything we need to know and do.... Rather, our personal responsibility is to learn what we should learn..."
This directly relates to teaching our students. If our students expect us to teach and tell them everything that they need to know and do, they will never actually engage in mathematics. We have to instead help them to engage in the mathematics and take personal responsibility to learn. We will still help and guide and direct our students in their learning process, but if we can help them take responsibility they will be much more successful.
As a similar note, students can't just learn from a single source. If they learn all their mathematics from their teachers they will lose the ability to do math as soon as they forget the teacher. If they learn all their mathematics from a textbook they will lose the math once they set down their book. If a student learns all their mathematics from exploration alone they won't get very far (it's taken us thousands of years to get where we are mathematically, we can't expect our students to get there in a few years by themselves). If students rely too heavily on technology, they will lose the ability to do math when their phones are taken away. Students need to have many sources and internalize mathematics from all the rich opportunities they have available to them.

"If all you or I know about Jesus Christ and His restored gospel is what other people teach or tell us, then the foundation of our testimony of Him and His glorious latter-day work is built upon sand. We cannot rely exclusively upon or borrow gospel light and knowledge from other people—even those whom we love and trust."
This is in continuation of the previous idea, if our students only know what we tell them about mathematics, they will be on extremely shaky ground. It is important that we teach and help our students build their own understanding rather than relying completely on ours. It is important that our students can trust that we know the mathematics, but if that is all that they will ever have they are going to fall and their knowledge will erode away over time (just as a building built on sand would do). Students who build their own ideas of mathematics and can prove or justify them by themselves will keep these foundational experiences and truly engage in mathematical processes.

"A rich reservoir of resources exists in print, audio, video, and other formats to help us learn about [temple ordinances]..."
While this quote was specifically about the temple, this is very true for our math students. They have a huge variety of resources to learn from, including all the dynamic software and other technology we talk so much about. If we are familiar with these resources, we can point our students in the right direction so they can also have a taste of all the technology available for mathematical exploration.

Well, that's about enough for my perspective. If it wasn't the end of the semester and finals coming up, I might have explored a few more talks, but hopefully that is sufficient for now. And maybe you, my fine reader, can explore and dive into some other talks.

Anyway, a little about the discussion that I had with my fellow educators about General Conference: We focused a little more on the technological aspect of the conference. Something that Elder Cook said is that we should make technology "a servant instead of a distraction or, even worse, a master." This definitely applies to our classrooms. If technology is a distraction or a master of our mathematics we have failed as a teacher. This is true whether the technology is a distraction to us or to our students. We must use the technology to our benefit, not as a deterrent to learning.

There is a paper by Vince Geiger that discusses the use of technology in a math classroom that is very similar in the wording. Basically, Geiger describes a series of levels of using technology in the math classroom, beginning from the least useful to most: Technology as Master, Technology as Servant, Technology as Partner, Technology as an Extension of Self. We discussed this article and tried to determine where we were with various technologies. As a whole, we came to the conclusion that with most technologies, we are definitely to the point of servant, and hopefully to partner and as an extension of self. As teachers, we need to be familiar enough with the technology to be at a place where we can help our students not be servants to the technology but be able to use it to explore and further understanding.

We then furthered our discussion a little bit about how to get technologies in our classroom. We had the perspective of a professor who taught for many years in public schools before transitioning to teaching in college, and that was very interesting. He talked about the many types of grants that we as teachers can apply for, how we could get new technology gradually, how to help parents and administration be on our side, and how to show the benefits of new technology. Overall this was very interesting, and very useful. If in our future classrooms we have no technology we are actually slaves to that lack of technology, which is almost as bad as being slaves to the technology. 

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.

Using Desmos in a Classroom Environment

I've recently been exploring online software for use in a classroom. While I was familiar with using GeoGebra for a variety of geometry and graphical representations, I was amazed with the other applications of Desmos as a powerful resource for teachers and students to explore. Apparently, Desmos has built a wide variety of classroom explorations. These are actually fairly high quality explorations, and I am actually going to continue exploring Desmos in a few explorations in the near future, so keep your eyes open for those.

In 2016 NCTM published an article about increasing class participation through online resources. While it was written by authors of Desmos curriculum, I feel like it gives great information and suggestions for math teachers. It's a little outdated, one of the activities that they describe has lost native Desmos software support, but the principles are pretty useful to think about. The article can be found here (or here if you have a BYU Learning Suite account).

Summary

This article, it seems, was written primarily to discuss the benefits of using online technologies in a math classroom. That being the case, one major benefits of using online softwares (particularly Desmos) in a classroom is that students can receive iterative feedback- feedback that comes extremely fast and not too leading. Online tools are also useful in that students can share with each other in additional ways, making student collaboration and peer to peer learning much more likely. Beyond just student feedback, teachers also have more information, often being able to observe the entire process of learning instead of just viewing the final result. The article then explores two Desmos activities and how they achieve these aims of online software tools.

As I discussed this with a few of my fellow students, we were amazed with the tools that are available for the teacher as students accomplish these tasks. This eventually progressed past a discussion of the article, and I will actually discuss these benefits in an upcoming blog post. However, we also discussed how online tasks should not and cannot replace classroom discussion led by a teacher. While student's interaction and peer teaching is extremely useful, we must also be able to "steer" the class in the correct direction, keeping students from "guessing and checking" when they do not understand the material. This is actually possible in any Desmos activity: the teacher has the control to "pause" all their student's screens, inserting teaching and discussion where otherwise students might get caught in a learning trap.

Critique

While this was an interesting article, I feel that it was much too focused on the benefits and value of Desmos. No other software was mentioned in the article, while I know that there are a variety of other softwares and online mathematical tools for the classroom. The article itself also didn't mention the need for heavy teacher involvement in a classroom full of devices, it almost seemed to me that the article was implying that the class would teach itself with Desmos. This makes sense as it was written by some authors of Desmos, but it was lacking in power by not addressing all parts of this important issue.

I do wonder about the discussion that we had and if we are all a bit naive. While our discussion focused on the ideal classroom environment, I do wonder how long it will take us to actually come anywhere near that in our teaching. Teachers who have been teaching for years may be able to know where each of their 200 students are in their understanding and address needs well, but I worry about being able to do that in my classroom. However, I do feel like online activities will give us additional information on student understanding and the excitement that our discussion had for the tools of Desmos is deserving. Being able to pause an activity is an extremely powerful tool, and recognizing that was a great benefit from our discussion.

Connections

I think that technology can be extremely useful in the mathematics classroom. For example, when I was in high school, I programmed my graphing calculator to solve a quadratic equation using the quadratic formula. I knew how to plug numbers into an equation, I'd done that hundreds of times and probably will a few thousand times more. However, for ease, I used my calculator and would get onto more "interesting" mathematics instead of wasting my time with something that I already understood.

The other day, I was observing in a classroom and watched as a teacher had students graph (on their graphing calculators) an exponential function versus its' derivative. Students then used that to estimate what the formula for the derivative of an exponential function from that graphical representation. This could have been very useful for these students, but I feel like it wasn't presented in an exploratory manner, making it a "plug and chug" method for many students. I think if it had been presented in a way such as through Desmos or other exploratory software students could have discovered and understood the concepts a lot better than they did.

Why Are Teachers Held to a Higher Standard?

Why Are Teachers Held to a Higher Standard?

Something interesting has been on my mind recently. I read news stories of teachers getting into trouble for a variety of reasons. All you have to do is to search "teacher firing" on google news and you can find hundreds of stories in the past month about teachers being fired. Some definitely deserve to be fired (a teacher having a sexual relationship with a student should be sent to jail for a long, long time in my opinion) but some seem a little too far. Here, here, here, here, and here are just a few examples of teachers being fired for a variety of reasons. [Note that I haven't done extensive research on any of these cases, I just present them as examples of the variety of reasons teachers can be fired.]

In a textbook about technology for teachers, there is a very interesting article about how teachers must maintain high standards with their online presence. The article can be found at this link.

Summary

In a nutshell, this article is directing teachers (and prospective teachers) to be extremely cautious with their online presence. Teachers must abide by rules from a variety of sources including national and state laws, district policies, and institutional policies. These include a variety of different categories: legal requirements, social requirements, and to avoid any "inappropriate behavior". The article points out that the teacher has no control over what is viewed as inappropriate, but that the school district, community, and individuals can judge what is "moral" and deem a teacher's actions to be worthy of firing even if the teacher may not think that what they have done is wrong in any way.

As I discussed this with a few of my fellow students, the conversation tended toward how we as teachers can exercise a little control over our online presence and limit the danger of our online presence being deemed inappropriate. It seemed to be the feeling that it is our duty as teachers to maintain a high standard of conduct. We discussed the need to keep social media private, but also to monitor and not post anything that may be taken incorrectly. This includes strong political or religious posts as well as any context of a sexual or sensitive nature.

Critique

While I agree with the need to be professional in all aspects of life, I do wonder why teachers are held to a higher standard than many other people. Even politicians, whose every move is scrutinized, are extremely unlikely to be fired if they correct someone's spelling on Twitter. In my mind, I believe that teachers are held to a higher standard because we are teaching the children of the future. However, I do find it strange that the parents of our students can hold us as higher moral authorities for their children. I feel like that has the possibility of creating confusion for our students and more than any one article or blog post could address.

Connections

Something that I thought about as I was reading the article was when I was a missionary for the Church of Jesus Christ of Latter-Day Saints. I was among a group of missionaries who was given access to an iPad and other technology in order to teach the gospel of Jesus Christ. Before we were given these iPads, we had a few meetings and trainings that we had to attend. We even had a book that emphasized some safeguards using technology. While this was specifically directed toward spiritual protection, we also learned a few things about how to maintain an online profile in a positive way. These tips were very similar to those tips that can be found in the article linked above. 

I tend to have a fairly small online presence. In fact, this blog is the most active I've ever been online. However, I've always been fairly conscious of what I post, likely because I grew up with a teacher and I remember listening to stories about people (specifically students and teachers) who shared too much information and ended up with their house vandalized. I also tend to be a private person, and my online presence very much reflects that. If you try to find me on any social platform, my profile will be relatively private. 

It's very clear to me that teachers are held to a higher standard than people in many (perhaps even most) other careers. While I considered titling this post "Are Teachers Held to a Higher Standard?", the answer to that is extremely obvious. I thus chose to title this post with a "Why" instead, even though I don't have all the answers. 

Making Technology a Tool

What teacher hasn't had to worry about how technology is being used in their classroom? We worry about students surfing Facebook, using calculators for basic addition, and even have to think about the capabilities of a student's glasses. So what do we do about technology in the classroom?

Recently I came together with a group of math educators and discussed an article about technology use in a spiritual environment. The article we discussed can be found here. It's well worth a read for any teacher; not just for those who teach in a church environment, but for anyone in a classroom.

Summary

The article describes a few different approaches that teachers can take to improve classes by using technology. Teachers are encouraged to use technology in a variety of righteous ways, both personally and as a teacher. In order to do so, we must be familiar with the technology and make it a part of our lessons. The article also points out that our students are more likely to use their technology in productive ways when we justify rules about technology in ways that do not critique the technology itself but instead focus on the reasoning behind the rules. However, the article does make certain to point out that technology cannot and should not replace face to face interaction between teacher and student.

As we discussed the article in our group, while we discussed the highlights of each section we seemed to focus on how to set rules and the process that we need to go through to learn about the technology. When we set rules in our classroom we should have justifiable reasons for setting those rules. Trying to force our students to obey an arbitrary rule may only encourage our students to rebel. This is especially true with technology since our students are likely to be more familiar with the technology than we are. At the same time, as teachers we need to do what we can to be familiar with technology and the tools available to us. By doing so we can improve our teaching and supplement our interactions with technology.

Critique

During our discussion, I realized how important it is that we keep abreast of the technology. It was pointed out that many years ago teachers were worried about students using erasers. Apparently teachers thought that writing would degrade because students could now erase their mistakes. This was clearly a mistake, pencils and erasers are extremely useful in the classroom. We must not make the same mistake in our classrooms. Instead of fighting a fight we can't win, we should figure out how to use new technologies to help our students learn. 

Connections

Something that was very interesting to me about the article and discussion was the importance of setting principle-based expectations. In my mind, I see students being much more willing to follow the rules of our classroom when they know the reasons behind them. This is a big part of adolescence, and our students want to be respected and treated as adults. Setting rules and explaining the "why" behind them allows our students to have autonomy in their behavior. This is something that is key to my personal teaching philosophy, and something that I wish was the case when I was a student.

When I was in high school, I was very involved in learning about technology. I actually explored how to use a graphing calculator and computers to generate and analyze prime numbers (see Golbach's Conjecture). None of my teachers knew how to program with a calculator, and I could have used that to do a variety of things. However, because I was so engaged with classes and other explorations with mathematics, I didn't ever make trouble with my knowledge. I believe that it will be the same with my future students if I can help them be engaged in the mathematics. Technology will be a tool for our students instead of a crutch or distraction.