Methods or Principles?

When it comes to solving a math problem, there are several methods a student can implement that will yield the correct answer. Sometimes, while helping friends with their math problems, I like to wait until they finish the problem to confirm their answer matches mine. This approach gives allowance to the fact that we learned two different methods for solving that sort of problem. However, because I grasp the general principle that they are applying, I can often double-check their work even if the specific notation or order of steps is different than what I utilized.

I have noticed that my understanding of general principles is not shared by all math students. This makes me question whether math teachers are teaching methods or principles to their students in the classroom. Many students struggle to solve a problem unless they are using the specific method they learned in the past because they do not understand the underlying principle behind the method.

“As to methods, there may be a million and then some, but principles are few. The man who grasps principles can successfully select his own methods. The man who tries methods, ignoring principles, is sure to have trouble.” -Harrington Emerson

Although this quote is usually applied to life principles, I find that it is just as relatable for a mathematics classroom, while also addressing the very observations I have made. Math teachers need to be teaching their students the principles and big ideas of a lesson. If a student understands what they are trying to accomplish and the principles behind finding the solution, they can then choose the method for solving that makes the most sense in their mind. Students are often taught multiple methods to solve a problem, but without an understanding of the foundational principles of the lesson, they will struggle to know which methods to use, why they are using that particular method, and when to use it.

A competent teacher helps their students understand the principles of math so that they can choose a method that makes sense to them. Teachers serve their students well when they expose their students to one main principle and multiple methods that address that principle. This idea is utilized in both lower and higher-level math classes. In Calc 2, I learned many methods for solving integrals. However, I first needed to grasp the principle of integration if I was going to be successful in choosing which method to apply to a problem.

Gaining a deep understanding of mathematical principles also means that students will be able to solve many different problems. If all a student knows is a method for solving a given problem, they will be confused when they run across an unfamiliar problem even if it uses the same basic principle. However, if the student has been taught the mathematical principle, they will have the foundational knowledge needed to solve problems that initially look unfamiliar. Methods are often easier to teach, but teaching principles are what will serve students best as they move forward in the world of mathematics. Teaching principles allows students to become problem solvers, not merely method masters.

Is Math Just One Subject?

Earlier this week I had the privilege of attending the South Dakota STEM (Science, Technology, Engineering, and Mathematics) Education conference in Huron, SD. I attended the conference with an open mindset of learning a lot of new things, however, one session, “Teaching Math as a Language,” left me questioning.  

The way the session opened was by comparing a math classroom to a language classroom. When he played the video of the language classroom, we were all a bit confused, as we had no idea what was going on. Later in the session, he presented an equation in slope form with a range of x-values. He then asked for volunteers to read the equation aloud. Two volunteers read the equation differently. At this point, I was confused about why or how they could read the equation differently.  

As a math education major, I adapted to the speech of math early on in life, but just like language classes, some people cannot adapt or process the language of math as easily. Translating math into language is a lot harder for kids; I have some younger siblings and I asked them to translate an equation for me. They also had three separate answers when reading the equation aloud (for reference there is a 7th grader, a sophomore, and a senior in high school).  

After witnessing this event happen two times in the same month with quite different people, I am curious how often kids struggle with translating a mathematical question and if they are struggling with just the basics of translation. Are they able to fully understand the concept of math? What if we started teaching the basics of how to pronounce equations in our lessons? How would this affect learning and the classrooms? Would this be beneficial, or would it take too much time out of the lesson and be too confusing?  

Referring to the session I attended, the presenter ended it with how he was teaching a lesson on chalkboard and one student was not writing anything down for the entire lesson. When he dismissed the class, he asked the student, “Why did you not write anything until the end of the lesson?” The student then responded, “I was trying to figure out how you wrote the curly bracket at the beginning of the lesson.” This brings us back to whether math should be taught as just a language or as a writing classroom. Is math just one subject? Do we need to teach math as multiple subjects so students can fully understand it?  

Virtual Reality in the Classroom

With Apple’s release of the Apple Vision Pro, virtual reality is becoming more prevalent daily.  With the seamless cross of the real and the virtual, the applications are only limited to one's imagination. This led me to research different ways virtual reality technology can be used in the classroom.

One common use for virtual reality in the classroom is a tour of a virtual museum.  Instead of having to plan, schedule, and attend a field trip to a museum students can now visit any museum available in VR.  This comes with some drawbacks though as some may think that students wouldn’t be getting the full experience of visiting a museum.  I couldn’t agree more.  There is a time and a place for VR tours, but you can’t beat going to a real museum in person.

Another use of VR in the classroom is to allow students to experiment with science in a safe environment.  Students of any age can conduct VR simulations of physics, chemistry, and engineering, without the dangers that come with the field.  I think this is a great application of VR in the classroom, as it allows students of all ages to experience the sciences in a safe environment.  For example, we could have elementary schoolers, doing simulations of chemistry experiments, without the dangers that come with bunsen burners and corrosive chemicals.  Or they could start trying electrical and mechanical engineering without the worries of them injuring themselves.

One more use of VR technology in the classroom is VR art.  With apps that allow you to paint, draw, and sculpt in VR, the possibilities are endless.  As the price of art supplies is ever-increasing, one would only have to have a one-time purchase of a VR headset to produce digital art.  Although some may view this as more graphic design than art, I believe it has enough force to become a medium of its own.

Even SDSU is hopping on the VR bandwagon.  In an article from March of 2023, SDSU was elected as one of seven universities to become a “metaversity.”  Students will be able to attend classes through a virtual replica of the SDSU campus.  They would be able to engage in the classroom just like a normal student.  This alone is a huge application for distance learning situations.  Whether you live out of state or are sick, you can attend your lectures like you were there in person.  If you would like to read more about it, here is the SDSU news article:

https://www.sdstate.edu/news/2023/03/sdsu-selected-one-seven-original-metaversities

As well, on February 28th, SDSU Faculty and Staff can attend an event that will teach them how to introduce VR in the classroom, if any faculty/staff would like to sign up, here is the link to the event:

https://www.sdstate.edu/events/2024/02/how-start-teaching-vr

With VR as an ever-increasing medium, it is only a matter of time before it makes its way into the classroom, and when it comes, I will welcome it with open arms.

Periodic table: Paper vs Phone App

The periodic table is a very useful tool when it comes to science classes, but when it comes to retaining all information about the 100+ different elements on the periodic table, some would find it to be quite challenging. Printing the periodic table out and customizing it to fit your needs is not the worst idea but wouldn’t it be easier to just download an app on your phone. If it is downloaded on your phone not only, can you take it everywhere you go but you don’t have to carry around a piece of paper. Trying to carry the periodic table around on paper can be just as challenging as trying to remember all the information on the table.

Comparing the paper to an app would make it quite easy to figure out which one you should choose. For starters, the paper copy and the app do share some similarities. Of course, they share the fact that the table is there for you to look at always and you can customize the table to fit the class. When I talk about customizing the table, I mean writing or even coloring the groups of s, d, p, and f, along with writing down the charges at the top of the table, and even grouping the metal, metalloids, and gasses. Along with all of these customizations, the only thing that you have to worry about is making sure that the paper stays in good condition, unlike a phone where you have to make sure stays charged and does not run out of battery.

The apps for periodic tables have the advantage over that on paper for multiple reasons. Yes, you have to make sure your phone doesn’t die but how often does a person let that happen. If you do your work at home, you can always keep your phone plugged in while you work. The periodic tables on apps also have most of if not all the information you need already on the app, it just comes down to you looking over the information and making sure you are prepared. The app also has some added information that can’t fit on a normal periodic table. Apps include the Latin version of the name along with the year that the element was discovered. That’s not all either, the app also includes the density of the element along with its melting and boiling point. Everything that could be found in a book or on the periodic table inside of a book can be found on one app on a phone.

In all, having a periodic table app on your phone would be just as valuable as having it on a piece of paper or trying to look at one in a book. The apps end up providing as much information as the periodic table and the book combined. People today would rather have things on their phones that they deem valuable and for chemistry students, the periodic table is very valuable.