It’s Not ‘What’ You Do, It’s ‘How’ You Do It

Back when I used to play in my High School’s marching band, my director would say this to us often. Now, whenever I describe the importance of band to others, I always include this saying. Yes, when you’re older, knowing scales, standard step-sizes, or alternate fingerings is not really important (that is, unless that’s you’re job). However, being a part of marching band is so much more than what marching band is on the surface. Younger students  Ironically, if you ask students why they are in marching band, they probably won’t mention marching or music.

Imagine my surprise when this phrase comes up again, this time in reference to learning from Harry Potter. In addition, learning music is an easy environment to observe an example of mindless overlearning. So then, I absolutely found this week’s reading particularly fascinating, especially Langer’s The Power of Mindful Learning.

Facts and truth are important, yes, however learning is more than just the information. It also includes how you process information. I think that is something we lack in our education. We shouldn’t stop at “This is true”. That’s where rote memorization stops. We need to expand; think about other questions. “Why is this true?” “Why is this not true?” “Can this be false?” “When is this false?”  You’re learning information, but not learning how to think.

Well, let’s tie this back to mathematics again! One theme of mindful learning is valuing the uncertainty of information. As a mathematician, that’s a bit difficult isn’t it? “2 plus 2 is 4”. “Closed and bounded implies compact.” Mathematics seems to be built upon immovable theorems and unyielding truths. While it’s true when Langer said “one plus one does not equal two in all number systems”, you can’t escape the fact that mathematicians pride themselves with making proofs that are absolute.

As much as I love mathematics, I envy the… “malleable” nature of other fields. If you study Foreign Affairs, a single news story can change the context of a class you’ve been preparing all summer for. There are new interpretations of literary classics that have been around for decades. Last class, I described mathematics as “dead” knowledge to my group. That is, it’s just… there. In contrast, something like history is “alive”. You can debate about different historical perspectives and implications; contrasting ideas don’t have to be mutually exclusive.

Mathematics is a blatant culprit of mindless learning. I’m surprised Langer doesn’t bash on us more in the first two chapters.

So can we be mindful when we teach math? Well, YES! Thinking back, I’ve witnessed the effect of mindful & mindless learning when it comes to mathematics. A particularly clear example is teaching integrals at the Math Empo (I worked there for two and a half semesters). There are so many integration rules students learn. They go to the Math Empo and grind away at practice problems, almost to the point of overlearning (!). I often hear “I’ll keep doing them until I don’t see any new integrals”. Color me shocked when I also hear “It’s not fair, the quiz had an integral that wasn’t in the practice problems”. When math teachers focus so much on teaching the rules we lose out on the “thinking process” of the integrals. If we don’t practice mindful learning, of course we will have students fail to apply math skills to new problems.

As math teachers, we should also be expressing the problem solving strategies that we think of as we go through a problem. Why do students have trouble with word problems? Because we aren’t teaching them in a mindful way. We teach the equations, but not how to think between the sentences and the equations. When you write a theorem down, think about how the proof doesn’t work if you are missing a hypothesis. When students ask you a question in office hours, don’t just tell them the answer, lead them there! When people ask tutors at the Math Empo for help, we (the tutors) always ask questions back. People complain all the time. Yet, that’s mindful learning. We are trying to have the student engage with math themselves.

Mathematics is about problem solving, not solved problems. When we focus so much on rote memorization, you lose out on the bigger picture. So again, let’s end with our favorite saying:

It’s not about what you do, it’s about how you do it. 

Environments & Education

Last week, a reading referenced George Kuh’s idea of experiential learning. Where learning can mean something more than just memorization; instead, learning is like an adventure.  A common thread between this description of learning and the examples presented in A New Culture of Learningis the idea that learning should be more holistic. It shouldn’t just stop at simply learning facts by rote and regurgitating them back at a teacher.

There is discussion on shifting away from lecture, and instead, focusing on creating an environment where students are free to explore and interact on their own to learn. I believe that when the initiative to explore (and learn) is given to the students, they become more engaged and in turn, more invested in what they are experiencing. When it works, this is a powerful technique to teach students, not only information, but the process in which they obtain it.

In this environment, failure is not only encouraged, but it is required to explore the boundaries and constraints of the environment students are placed in. The ability to reiterate and experiment without the fear of failure is natural learning at its finest. And it is through that failure where students begin to innovate.

But that’s the trick, how do we make it work? In certain contexts, it is clear that this form of teaching is better for students. You can tell people what happened in the past, or you can design scenarios where students live through it themselves. However again, I’m thinking about how this connects back to teaching mathematics. Can we create that environment of exploration when it comes to higher level mathematics?

I took MATH 3114, Linear Algebra, with Professor Wawro my first year at Tech. Professor Wawro does research in Math Education, so unsurprisingly, her class was not a typical. She set up the topic of diagonalization in a way where we almost “stumbled” upon it by our own exploration. We were presented with a problem before us, and through the process of solving it, we unknowingly described the technique for the change-of-basis matrix. Looking back, this was really the only example of “exploration” in teaching mathematics that I have experienced.

Now, I think it is important to see the strengths of lecturing as a technique as well. I was very glad to see this article about someone who, while not a full proponent of lectures, still finds lectures helpful in certain ways. I really appreciate someone acknowledging both sides to the argument. I’m not sure about fully replacing current techniques with these new ideas, but rather, to use new techniques to support the classic techniques that we use.

Perhaps because I’m so focused on thinking about how to teach mathematics that I, myself, am missing that bigger picture, the holistic framework of learning.

My final thought is on the connection between video games and learning. Reading about  learning theories in video games brings reminds me of a youtube channel I stumbled upon two years ago called Extra Credits. Among other game-related topics, they created videos discussing games in education. Some topics include gamification of educationagency in education, and also include some small case studies. While their videos aren’t necessarily deep, they are an interesting watch for those who have dabbled in video games in the past.

All this Contemporary Pedagogy… Why should I care?

As my first foray into blogging, I apologize in advance for my lack of writing skills. As someone who sits in the STEM field (Mathematics to be precise), it is critical to practice communication skills. However at first, my writing is going to suck. Watching the TEDx Talk by Wesch, I have to acknowledge that failure is a part of learning.

By taking this course in Contemporary Pedagogy, I am placing myself outside of my comfort zone. When I talk about what this course to my peers in my department, I always get a confused look. Integrals, Analysis, Computation, most of my peers only focus on the math. Blogging, group activities, portfolios, what do these have to do with a math class? Can there really be another way to teach definitions and theorems? I’m not sure. By taking this course, I hope to expand my perspectives. Perhaps there’s a way to change how we teach both math classes focused on rote exercise memorization and classes focused on definitions and theorems.

This week was labeled “Networked Learning”. Specifically, our readings focused on the power of using the internet to connect those who seek to learn. Specifically, one online tool that enhances academic research is blogging. I feel that it is pretty clear that blogging has many benefits for the humanities and the social sciences. The previous link referenced many blogs of that type. But can blogs be a useful tool for mathematicians as well? Why, yes! The creator of the ubiquitous programming language, MATLAB, has a regularly updated blog about numerical programming. It’s easy to think some practices only work well in one field or another. I say, that’s just being too lazy to figure out how to adapt practices in a novel way.

My last thoughts for this week comes from the article about networked learning. What struck me specifically was the idea that the library is one massive example of networked learning. We’ve been doing this ever since the creation of the written word. The point is though, the tools that we use to connect to each other are changing. The printing press changed the way people learned in the past. Today’s “printing press” is the internet. And so, it is so very critical to learn how to use networked learning to further our knowledge.

Again, I always try to think about how networked learning applies to my field, mathematics. This one is surprisingly easy. Stack Exchange is a huge platform for users to ask questions, and answer them as well. You see questions about first year calculus, and programming logic, all the way to graduate level information is some obscure mathematical field. Personally, it’s amazing to witness such a massive example of networked learning. Perhaps it’s not a stretch to see how Contemporary Pedagogy applies to mathematics.

The Journey Begins

Wow! This will be the first time I’ve ever handled a blog before. While it is for a class, I think it’s important to learn the communication skills required to keep up with a blog.

Good company in a journey makes the way seem shorter. — Izaak Walton