Stream of Consciousness

Some Thoughts on the Practicality of Mathematics

by Cacey L. Wells 6/9/2022

I think there is a common misunderstanding when it comes to teaching mathematics. That is, a lot of folks I talked to think teaching math should consist of crunching numbers, memorizing formulae to crunch said numbers, and using said formulae and numbers to do something tangible to one's everyday life. I find this notion to be neither practical nor realistic. How often are your crunching numbers or applying formulas in your everyday life? Probably not much, right? However, how often does being able to problesolve a situation arise? Probably multiple times per day. Think about it for a second. You probably had to do some sort of problem solving in order to navigate your way to this webpage or packing your kids' lunches before they left for school.

In Everyday Calculus, Fernandez walks his readers through a typical day and showcases how calculus plays a role or how it is surfaces in what we might consider to be mundane situations. What I didn't find in this book, though, was a worksheet where he shows you how to calculate integrals, but examples of how mathematics explains situations. As I read this book, I came away with a better sense of how calculus undergirds most of our lives. It isn't that I need to know how to solve differential equations, although these are sometimes fun little puzzles to explore. That being said, our everyday lives don't necessarily require us to simply crunch numbers, but to think. To think deeply, at that. We should know how to make simple calculations, but what if your math class consisted of an appreciation for mathematics with opportunities to see how it manifests itself in your walk around the neighborhood or in the architectural choices made to construct the beautiful building you drive past on your way to the office? That seems more practical to me than being able to determine the number of legs on the beach if there are fifteen humans, eight octopi, and twelve dogs.

Authentic Mathematics

by Cacey L. Wells 12/15/21

A few years ago I was working as a professional development coordinator in graduate school at the University of Oklahoma. Part of my job was to work with teachers to help them implement what we called "authentic" teaching and learning practices into their classrooms. Most teachers I worked with were located in an urban school district facing many challenges that one might assume you'd find in high school settings like that. Primarily lower income families, racially diverse, emergent bilingual populations, mostly white teachers. There seemed to be a disconnect between the lived experiences of students and families and the experiences of teachers and administrators. Many teachers would drive in from the suburbs and could escape to their separate communities. This wasn't wrong, but the fact that teachers weren't part of the school community made the experiences in the classroom feel less authentic than they might be if teachers had been part of the community and had similar life experiences. Of course this wasn't always the case, but was true of many teachers in two schools in particular.

All of that is to preface the fact that I came into the school as a clear outsider to watch teachers in their classrooms with kids that mostly didn't look or speak like them teach a lesson that was supposed to be authentic in nature. The lessons chosen by teachers were created by people like me – in a vacuum at a university and not in context of the school itself. I'm sure you can guess how many of these observations went. Teachers rarely taught the lessons to a high level of fidelity. Or, if they did teach the lesson close to how it was intended, the lessons would often fall flat and feel less than genuine. Part of this had to do with the fact that many lessons were created with a constructivist or social constructivist ideology in mind – meaning these lessons were intended to be hands-on explorations with little lecturing or top-down instruction from the teacher. As you can imagine, this pedagogical approach was in pretty stark contrast to how many students were conditioned to learn. In a low-performing school, there is an immense pressure to perform better year after year. Since funding is tied to outcomes of standardized tests, this puts tremendous pressure of teachers to have their students do well on end of year exams. The result often looks like teaching to the test, focusing primarily on standards, and training students with problems to solve that are in line with standardized tests. Couple that pressure with behavior issues found in schools, sometimes unsteady lives outside of the classroom, language barriers, food insecurity, and a myriad of other factors not taken into consideration by those who make these exams, and you'll see why report card data tends to favor wealthier, white communities.

So, what is authentic instruction and how could these lessons have been made to be more authentic? In the mid-1990s, Fred Newmann and his co-authors coined the term "authenticity" as it relates to teaching and learning. Essentially, this boiled down to three primary tenets: 1. Students have opportunities to construct meaning for themselves; 2. Students are engaged in what they called disciplined inquiry. This consists of asking meaningful questions and engaging in substantive conversations, and; 3. Students make real-world connections. I think these are all good ideas, but may be lacking a little in terms of how authentic teaching and learning manifests itself in school. The third tenet seems to be problematic since "real-world" doesn't necessarily encapsulate the students' world or lived experience and implies that the subject matter should be useful to a profession. I like to amend Newmann's third tenet to say "connections" since these real-world scenarios may not take into consideration a student's interests or lived experiences.

Further, I think the fact that the classroom environment is only hinted at in these three tenets is also potentially problematic. If the classroom is structured in a top-down, hierarchical way, a teacher's ability to lean into a constructivist ideology is nearly impossible since this pedagogical approach is diametrically opposed to a more traditional teaching model. Moreover, if students have not had opportunities to converse in class, share ideas in a collaborative way, or engage in substantive conversations with classmates, they may not know what to do when given this opportunity. It's like throwing them into a foreign environment without the appropriate apparati to succeed. Classroom environments and how these are created/cultivated are vital to the success of authentic instruction since they frame the situation in which students find themselve.

Finally, finding relevant content to help students explore within this framework is key. If classroom content is created by a textbook or a teacher who isn't fully situated in the school culture, the explorations with which students engage are low on meaning/interest. If students and teachers can work collaboratively to come up with problems they are interested in solving, these can be much more authentic and engaging for students to explore. This might bridge their curricula into project-based learning or experiential learning as well.

Why a Collective?

by Cacey L. Wells 12/1/21

It's no secret that I really enjoy mathematics and love teaching mathematics. But, I also like writing about mathematics, taking photographs that have a mathematical theme to them, and playing games that have a mathematical twist. These ideas don't always fit the mold we often associate with how mathematics is learned in the traditional sense. There's a lot of discovery involved in making sense of the world through a mathematical lens or walking around town intentionally looking for mathematics. I know I'm not alone in these sorts of endeavors and that's why I think it is important to create a space where mathematics enthusiasts can come together to share ideas, interesting mathematics finds, and other creative intersections.

While I'm currently the only member of the Mathematics Creative Collective, I'm hoping this space will grow and other will come together to share their creative mathematical endeavors. This Ideas page will serve as a home for this blog where I'll begin posting ideas and musings from my perspective. As this space grows, others will be able to contribute their thoughts.