After watching the video, take some time to think about when you yourself experienced a transformation in your life. Write your own threshold story and post it as a new topic in this discourse category and then respond to two other threshold stories shared.
You can respond however you see fit, but we recommend the following thinking routing:
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What values does this story invite us to think about?
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Who is this story speaking about? And who is this story trying to speak to? (We know its a personal story, but think about the “character” in the story before and after. Who is this character in the story, as it is told?)
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What actions might this story encourage?
In college, I took a math course with a Fields Medalist that was way over my head. I got little in the way of mathematical knowledge from the course, but one thing happened that fundamentally shifted my view of what mathematics is, how it is created/build, and who does the creating/building. In general, the course (algebraic geometry) was filled with lots of definitions for different types of mathematical objects. Some of these definitions made sense, while others seemed totally arbitrary and unnecessarily long (think of a page-long definition containing a number of different specifications and requirements). Over the course of the semester, I was building up a dictionary of terms and definitions that needed to essentially be memorized. One day, the professor came into class and began describing a new concept - dimension. Instead of jumping straight to the definition (which again was convoluted and long), he told the story of how the definition for dimension evolved over time. Prior to the 19th century, the commonly accepted definition for dimension was the number of coordinates needed to specify a point in space. So a line was one dimensional because you just need one number to specify a particular point, a plane was two dimensional because you need two coordinates to specify a point, and so on. He then described how Cantor, in the late 19th century, proved a one-to-one correspondence between the line and the plane, showing that for every point on the line, you can assign a unique point on the plane so that the entire plane is covered. The implication of this was that every point on the plane could be uniquely assigned one number. Given the original definition of dimension, this meant that the plane would be one dimensional. This led to a modification of the definition for dimension by the mathematical community. The professor described how additional insights then led to further modifications. I won’t get into all that here.
What I realized during that math class was that (1) mathematics is constructed, (2) it is constructed by humans, (3) it is constructed by humans working in communities, trying to solve problems and make sense of the world. This led to a paradigm shift away from a conception of mathematics as a perfect pre-packaged set of ideas developed by geniuses and made for me to consume, towards a conception of mathematics as a consciously set up system of knowledge prone to human failure and revision.
The first time I encountered a threshold in my studies is in the two ways physics classes are taught in high school and in college.
I’d taken an AP Physics C class which covers the introductory college physics course. The curriculum was roughly broken into units to study the important laws/equations such as the kinematics equations, energy conservation, electric field/charge, magnetism, and then closer to the end we started doing circuits with capacitors. As the goal of the class is to pass an AP test in the end, most of the learning objectives include knowing what equations to use in what circumstances, and to solve problems quickly and correctly. So this involved a lot of memorizing equations and doing lots and lots of practices. For the most part, problems are usually worded in such a way that it’s immediately clear which equation is meant to be used, and which unit this problem comes from.
MIT doesn’t take AP credits for technical courses, so I ended up having to take an introductory physics course again. However this time, instead of being told what equations “describe” which physical phenomenon, we began with the phenomenon and derived equations from it; the professor had also showed the different equations, such as the five kinematics equations or the different versions of Maxwell’s equations, are actually related to and derived from one another.
There were two threshold being broken here; first being that seeing materials being presented as how they were derived made me realize, similar to Milly, that physics isn’t just some archaic laws we take from a bible of truths, but a field populated with other curious scientists from past and present like myself. Secondly, I begin to see the interconnectedness of these concepts and equations in a way that’s more essential than seeing them as discrete circumstances.
My threshold moment came when I moved from rural Wisconsin to start college in the Milwaukee area. My first semester of college ever, I took a silly class that completely changed my perspective on education and helped me become a much more open minded person.
My first ever college class was a mandatory freshman seminar where I was learning about anthropology through the lens of Harry Potter. Silly as it sounds, we read the entire Harry Potter series over the course of the semester in conjunction with this little orange anthropology book that detailed what it means to be a human. Using that little orange book, we dissected not only the worldview and facets of humanity in the wizarding world, but also in our own. It was one of the first times I can remember being so explicitly challenged on my worldview by such a kind and diverse group of people. It was a life changing experience. I grew up in a rural town of 1000 people with a graduating class of 46 including me. I had known most of the 45 since kindergarten. I had never been challenged or exposed to so many different thoughts in my life as in this class about Harry Potter.
Through this experience I was able to recognize that the way I was raised, my history, my perspective were a drop in the bucket of possible ways to interpret the world. People from all over my state, the country, and some from overseas shared their own life stories and thoughts on the world. I remember it being challenging at first; I had never had my beliefs questioned or dissected so openly. But over the course of the class, I learned to speak less and listen more. I learned to not only be open to understanding other’s perspectives but also to changing my own. These are skills that have grown even more over time and have become essential to my everyday life.
One of the major threshold stories I can think of happened when I was in my fourth year teaching. I always wanted to make sure my students remained “responsible” in the past, so I took away points for late work and had points where I no longer accepted them. Then, one day, a colleague of mine asked me why. When I gave my reason, he said “Do you lose pay when you are late to work? Do you think you are a responsible person?” And it hit me. I was not helping my students; instead I was hindering them. So I changed. Instead of docking points for late work, I would have conversations with students about why they were late. It led to them sharing more if they were confused or uncertain or they admitted to something going on in life. Obviously, you still have the students who don’t work, no matter what, but I find that I have a new level of trust with students and they are more willing to work through their problems or come to me for help. It also allows them to try and fail because I also allow them to redo some assignments, so that they learn mastery of a skill.
That one conversation with my colleague sparked so many more changes to my philosophy and how I see grading. I am so appreciative that he took time to ask that question.
That sounds like a great course! I do miss that about college. I took many classes where some of my beliefs were questioned and stretched. I really wish I could back for some of that again. I am glad you took such a great experience away from it.
Growing up, I always wanted to learn about photography. I bought garage sale cameras that rarely functioned and used a handful of disposables that never quite worked out. Part of this was envy of my brother who had an old Canon SLR in his room that he was gifted for photography class in high school; he’s eight years older and camera was from an uncle we rarely saw.
When I finally hit high school I was certain that of the electives offered, I was going to take photography. With our strange split 9-10/11-12 structure in Minot, ND, I had to wait until my junior year. With my brother’s camera long gone, I picked up the same model the school was using (a Canon A-1 for those curious) cheap from an eBay auction.
I carried that camera with me everywhere. I was spending several evenings a month at local arts events at the time and I was doing my best to photograph fast-moving musicians in dimly lit spaces. I jumped into the deep end. And the photos were terrible — underexposed, blurry, often indecipherable. However, I knew there had to be a way to do better.
After a dozen rolls of guess-and-check frustration, I slowed down and did my best to analyze what was going wrong. I asked photographer friends for tips. I reached out to Mr. Serr my photography instructor. I used the internet to dig through websites of technical information. I chipped away modifying variables — film types, aperture settings, shutter speeds, off-camera lighting rigs — until I hit something that felt actually workable. I began to understand what was going wrong rather than just simmering in the disappointment of another failed roll of film.**
Everything changed when I finally internalized the connection between all of the elements of exposure: film speed, aperture, and shutter speed. When I realized that shifting any of these three variables effected the outcome of the photo — changing exposure, steadying the shot, increasing or decreasing the depth of field — I felt like I’d cracked a code. I still had a lifetime to hone my abilities, but I knew that I finally understood the core technical skills necessary to relax and enjoy the process.
I truly wish that I had spent more time in my youth studying mathematics. As a teacher, I am so often told that mathematics always has a “right and wrong answer” and I do my best to explain that it isn’t nearly that cut and dry. I may have to highly truncate your story and share it with some of my students.
It feels a bit like cheating to lean on the example provided by someone you know personally. However, it was student teaching under you that firmly established my resistance to penalizing late work. You presented the idea to me in such a way that I could not ethically rebut the idea (one which I was very familiar with from my own education). Thank you for that insight.