Dancing Euclidian Proofs

 I found it interesting how it took time to choreograph this dance, and how there was a trial and error aspect to it. At first the proofs did not seem to lend themselves to dance, but after changing some base assumptions of how the dance should look things fell into place seamlessly. This embodies the process that all mathematicians go through, where roadblocks keep being hit until we have an epiphany moment that changes everything. This makes me think that not only were they able to dance the proofs, but they unlocked a new way of doing the math. Similar to the invention of symbolic algebra over classic geometric. They felt that things were "off" when dancing improperly, perhaps by dance we could solve some of the stickier problems in math.

I enjoyed the discussion of metaphors. The critique of their bodies not exactly representing the mathematical constructs was something that crossed my mind while watching them. But they make excellent points how our visual representations of math objects is imperfect as well. When one comes to term with the facts that no matter what tools they use, they will be imperfect representations, we can include so many different types of representation. And with these new modes, we may gain new insights.

And I think this idea of imperfect representation is one that is the most applicable to the classroom. Educators are trying to convey knowledge to students, and use a myriad of methods to do so. The words in a lecture, the drawings on a board, the feel of 3D printed shapes; all of these are imperfect yet useful. It is ignorant of us to decide that some modes are more imperfect than others. If I want to say that I tried my best to convey an idea to my students, then I must use all of the tools at my disposal. 

Why Euclid is Beautiful

 The beauty in Euclid's work does not come from the math that he discovered. In fact, he discovered very little of the math. Instead, the beauty comes from the connections that he made between all known math at the time. One could call this the "meta-math." He was able to sort through postulates that were seemingly inconsequential to one another and see the structure beneath.

Euclid has been so important for millennia because his structure showed mathematicians how to rigorously prove their ideas. The result of one's theory is only as good as the assumptions one makes at the beginning. "Garbage in, Garbage out." But with Euclid's structure, math became safeguarded against garbage entering the system. This garbage-less system is the source of beauty in his works.

Another source of beauty comes from knowing how much can be done with so little. By making only five assumptions (and a few "common knowledge postulates") all of human mathematical knowledge was laid out on the table. This beauty is like the beauty of a tree's branches spreading out from its trunk.

The final source of beauty came far after his death, when the fifth postulate was removed. For after its removal, an entire new field of knowledge sprung into existence. Or perhaps it was already there waiting to be discovered. This miraculous event is in my mind equivalent to marveling at a tree, digging through the soil underneath, and popping out into another realm with trees of its own.

Beauty in a system is simplicity. Euclid became the first person to show how complex ideas are just made from simple statements, and how those statements have the power to change entire fields of math.

Babylonian Algebra

The algebraic notations we use today are new inventions that have enabled us to discover many things. However, one could argue that by implementing these symbols, we have lost sight of the true mathematics. Many students leave high school math thinking that math only exists in the symbolic form, and that anything that is not explicitly stated via the symbols is not math. The invention of these symbols has helped, but they has also become a sort of false idol.

The invention of i as a solution to a quadratic that has no real solutions is a perfect example of how symbolic algebra has allowed us to leave the constraints of our reality to understand math at a deeper level. Symbols allow us to see past our plane of existence to what logically does exist. Our senses no longer dictate our understanding of math.

But we have swung too far on this pendulum. Students now do not value their senses to perform maths. Math exists in front of our eyes yet we do not look for it. Most math in history was done geometrically with physical objects. And even the modern theories we have today are often best described using visuals. Even imaginary numbers themselves are easily conveyed as 2D numbers. Just because a proof involves our physical senses, does not make it any less worthy. 

It is up to us as educators to bring the perception of math back in harmony between our senses and our symbols.

Was Pythagoreans Chinese

 By framing a knowledge as your own, it creates a perceived superiority over others. During the cold war, the ability for the USSR to put a satellite into orbit did not rouse cheers from the Americans, but despair. Cultures often compete with each other over which is more advanced, and this is often measured by which has the greater academic knowledge. We are now living in a global culture, which means that there should be collaboration, not competition, between nations. If we continue the false narrative that knowledge was only developed by "our" culture, then we lose the ability to work together. In competition there is no sharing of resources between sides. If our global culture is to overcome the challenges faced in the upcoming years, we must work together and recognize that all cultures have knowledge to bring to the table.

I am torn about the naming conventions of certain theorems. On one hand, I recognize that calling it "Pythagorean Theorem" emphasizes a the competitive approach to knowledge I mentioned above. However, I do believe that if it is taught with the emphasis that it was not only discovered in western society we can acknowledge that a name is only a name. Many cultures have discovered the concept of a chair, and thus many cultures have different words/names for the concept. It would be problematic if western students were taught that only their culture were smart enough to invent chairs and that is why they call it chairs. Hence, calling it Pythagorean is not a problem as long as the concept is introduced through an international lens.

Two visual Proofs of Pythagoras

The first is a 6 minute video using triangles inscribed in a circle:
Proving the Pythagorean Theorem with a CIRCLE - YouTube

The second is a physical demonstration using water:
Pythagoras Theorem Practical Proof - YouTube

Assignment 1 Reflection:

 This presentation went incredibly well. Not only was I able to understand the fundamentals of my question, but I was able to use my pre-existing knowledge to devise an activity that showed why my historical concept was so important.

What pleased me the most about this assignment was that I could play to my strengths. Most of my peers are mathematicians, while I am only an engineer. At first I thought that this was going to be a major detriment to become a successful teacher. But I learned that I could use this knowledge to my advantage and play to my strengths. 

Engineering is applied math, and this assignment has raised my confidence. I will be able to bring a new perspective to my student's education. Though my engineering lens I will be able to show them relevance that traditional word problems could never attain.