When I began the semester in my History of Mathematics course I thought math meant, “The study of patterns in the world and in our minds and how they connect to each other”. I even created a blog post called “So when I say Math I mean…” sharing my thinking and my scarce, but meaningful prior knowledge about the most important discoveries in mathematics. My definition of mathematics and my understanding of where the field is at today has expanded by at least five times the amount of ideas that I thought mathematics contained before the semester.
I would say that the way I define mathematics has been refined by the course as the discovery of patterns in the world, in the minds of men and women, and how these patterns connect to each other and extend beyond the point of our human understanding. One update to my definition lies in the point that I now believe that mathematics is discovered while notation is invented to explain our discoveries in mathematics.
Additionally, I would say that mathematics extends beyond our understanding because as of now I think forever there will be quantities, shapes, and ideas that connect the way the world works that we cannot comprehend as finite beings. Through our discussions in the math capstone course, I have come to have a view that mathematics is so much bigger than I first thought because mathematics can explain social, physical, natural, and celestial phenomena to me it can be viewed as a seperate, but connected to science. For a more in depth discussion on how many sciences there are I read a neat post on the number of sciences. I think for me personally throughout my time in the history of mathematics, I have come to realize that I do not really care whether mathematics is a science or not? What is more important is that I understand that the field of mathematics when explained well and utilized to its fullest potential, makes science easier to understand. Mathematics allows for amazing calculations, thoughts, and ideas that inform the concepts of science, future scientific discoveries and research.
Throughout this semester I have had a number of my preconceptions about matheamtics challenged, and have been reminded of the beauty of mathematics. Of my misconceptions the one that has been most challenged was that mathematical figures rarely look really exciting and interesting. Also, I did not realize that mathematical objects could be so mobile. Tesselations in particular stood out to me as I remember briefly learning about them at some point in high school. I came to see tesselations value in helping students to recognize the awesomeness of mathematics. Students can see how we can discover tesselations that no one else has ever created, but were possible all along by using some deep thought and drawing a repeating pattern that students must make reasoning choices and possibly calculations to ensure that the shapes or figures they use connect perfectly. Students may blindly fit patterns together like putting together legos, and it is the role of a teacher to help students to make their thinking visible in regard to shape selection for their tesselations. Students in everyday life make choices which require mathematical reasoning without even noticing that they are mathematical. Teachers like myself, need to help students realize that doing mathematics is normal, and that they already use mathematical reasoning in their everyday lives anyways, so doing mathematics at school will serve to improve their reasoning and ability to make use of structures to solve everyday problems.
Another perception of mine that was further challenged was the idea that mathematicians sit at a desk and work at problems for hours on end, rarely getting up to do anything in the real world. Archimedes was a prime example of the falsehood behind my thought. Archimedes built a claw as seen below which picked ships up out of the water, and destroyed them by forcing them to capsize. This invention was considered one of the greatest defensive war mechanisms of his time between 200 and 300 B.C. He also created a death ray from mirrors that was said to set fire to ships. Archimedes made many other vital contributions to mathematics like the calculation of pie to many more digits than it had previously been discovered to, and the creation of His wheel. Some other amazing mathematician works that impacted their day greatly were Newton’s discovery of the laws of gravity, Gauss’s invention of the electromagnetic telegraph a huge step in long distance communication, and Mandelbrott’s work with Fractals which contributed to the concept of space filling curves now being used when considering traffic flow in cities.
The aspect of mathematics that most astounded me throughout the course and had a big impact on my view of the purpose of mathematics was the concept of topology. Topology is the the study of geometric properties and spatial relations unaffected by the continuous change of shape or size of figures(google). Topology allows mathematics to extend far beyond the obvious visible world, to allow people to find new solutions and study the world in unique ways because topology shows us that we can change shapes and figures to form brand new objects that do not seem possible from an initial view. A great example of this is in Topology playdough what appears to be playdough with two holes and one hole locked around a pole can be molded so that two holes can be around the pole. Another example of the application of topology can be in untying knots as topology shows us that there are many ways to untie a knot beyond just reversing the steps by which the knot was tied. Topology may lead to many of the next great discoveries in mathematics.
To wrap up, Mathematics is so much more than countable patterns, and stagnant shapes, but a language and framework by which we can better understand the world. Mathematics is exciting, meaningful, and necessary for unlocking the sciences. Mathematics is beautiful, and I did a lot of mathematics throughout the Math Capstone course, and came to the realization that mathematics is not always obvious to us when we are first forming patterns, or manipulating numbers, but that is why mathematics is discovered. Following the discovery of mathematics people invent notation to describe their discoveries. The end quote says it all,I have thoroughly enjoyed the history of mathematics and I hope that you can enjoy it too! Some great places to begin your search are…