Critical Pedagogy


This post was a collaborative effort with Jack Viere, Heather Corley, Allie Briggs, Bailey Houghtaling, Tami Amos, and Neda Moayerian

This past week, our class used the Jigsaw Classroom technique to explore the ideas of Paulo Freire and Critical Pedagogy. Through the discussion of our different sources, we identified common themes, topics, and key words. As a table, we define Critical Pedagogy as the following:

Critical Pedagogy first acknowledges power structures in order to reciprocally cultivate knowledge within a dynamic learning space that acknowledges varying human perspectives and life experiences, promotes continual questioning, and liberates marginalized view points.

When looking to consolidate our ideas together, we decided to make a mural, or a shared whiteboard. The featured image above shows the whiteboard as of Wednesday, 3/28. The whiteboard has various phrases, ideas, and themes that we felt were important to the idea of Critical Pedagogy.

While it is creative, it also ties into the idea of Critical Pedagogy as a contrast to the Banking Model of Education. Freire critiques the Banking Model, which views students as nothing more than vessels to accept the knowledge given to them. In view of Critical Pedagogy, both students and teachers have the opportunity to learn. In order to keep with that idea that learning involves both the author and the reader, we invite you to add your own contributions to our whiteboard!

The Universal Language… Not So Universal?

Mathematics is often touted as The Universal Language. Supposedly, despite the many different symbols, definitions, notations, and conventions, the core principles of mathematics transcend cultural, societal, and even terrestrial differences. However, a language requires someone to speak it. And here, we find our universal language is not quite the same for everyone.

Coming into this week’s reading, I’ve been aware of the disparities for women in STEM. It exists, and there are multiple aspects to the problem. Personally, it’s just intiutive for me to think that equality is good. “Equal pay for equal work.” That just seems so obvious me. However, for me, I’ve always thought of striving for equality for equality’s sake. What I mean by that, equality is a right, and discrimination is bad. We should be more diverse in STEM, because everyone should have the opportunity to enter this field. It’s a question about morality and fairness.

Now, discrimination and racism can be a detriment to a student’s learning experience (and I will discuss that later). However first, one of the readings this week challenged my viewpoint that diversity is just a moral prerogative. The article, How Diversity Makes Us Smarter, speaks to the benefit of diversity in problem solving. When inter-disciplinary teams and experiences lead to innovative solutions, why do we not place the same importance in social diversity? I truly have never thought about this. Homogeneity excourages complacency. Novel ideas and solutions come about through constant questioning, and I never considered that diversity can have an impact on that.

In one of our additional readings, Whistling Vivaldi, Steele discussed the challenges women face in mathematics, as well as his attempts to understand what exactly is the cause. I’ve also been curious to find a reason for the disparities (and there are many). It’s important to acknowledge issues like harrassment. Part of me had some gut feeling that there was some gender biases that add extra obstacles for women in STEM. I feel like this chapter of his book really hit the tip of the ice berg. Without the presence of directly prejudiced people, Steele found that women can still feel negative consquences of gender biases. In addition, the effort to fight that stigma, itself, can hinder their ability to achieve. In my opinion, this really hits the nail on the head.

In terms of the classroom, how do we fight against this societal pressure? I feel like we can try to remove the stigma of “not understanding” material. Personally, I have times where I hadnot grasped subjects at first glance. When I don’t understand, I feel a stigma of asking questions in front of my peers during class. “I don’t want to seem dumb in front of others.” However, that doesn’t help me learn; it does the opposite. That pressure of trying to prove to others (or to society) of your abilities, while it  affects many more women in mathematics, it is not a unique feeling. We should strive to lower that pressure among our students in mathematics. Perhaps change the notion that only people who don’t know the material go to office hours.

Now, gender disparities in mathematics are not the only challenge that we face in our field. After some thought, I was starting to think about how language plays a part as well. Despite how we always think of math as transcending language, when it comes to expressing ideas to others, we have to use words in some way. For a direct example, think about foreign students who take proof courses. Is it more difficult for students to learn proof structure when they have to simultaneously learn English sentence structure, and grammer? For those new to proofs, those lexical aspects are absolutely required to learn how to express mathematical aspects . Native speakers have the luxury to focus entirely on the proofs, while non-native speaks have an additional handicap. This was a small thought of mine, but it would be interesting to explore the challenges to mathematics due to language barriers. Have there been major advances that took years to confirm because of difficulties in translation of mathematical papers?

Overall, I feel like Inclusive Pedagogy is much more relevant to mathematics than I initially had thought.

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