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I am not a formal teacher or a mathematician, but a mechanical engineer who loves to learn and derives great satisfaction from mentoring other up-and-coming “S.T.E.M.” professionals/students. Due to my lack of educator credentials or particular knowledge on the subject, my response reflects personal views and experiences.

I was terribly bored by math during my primary and secondary education. I was much more prone to want to tinker with things to see how they worked. I used the knowledge that I gained to build and create things that I enjoyed or found useful. When the time came to select a major while filling out my university application I chose mechanical engineering because it would extend my understanding of the mechanical objects that had captured my fancy. I made this choice in spite of my mathematical insecurities because of my passion for the mechanical and my desire to understand it.

My first math and math based courses were extremely difficult for me, but I persevered and received good grades. However, my grades were more a result of my ability to regurgitate information effectively than a result of a sound understanding. Here and there a light would come on and things would make sense, but by far and large I was being led on a blindfolded path, copying the recipes that I was instructed to follow.

Through it all, I did feel very empowered as I learned that I could make, design, or mathematically model nearly anything that I wanted to if I could just find an equation in some textbook for the problem. This spurred me to develop a can-do attitude toward anything that I wanted to do. This new mental paradigm and tenacity was soon to be called to good use in awakening my understanding of what math really could be.

While studying heat transfer I was introduced to PDE’s. I had actually previously had a formal introduction in a class on differential equations (focusing of course on ODE’s), but the regurgitation education model had left me with virtually no recollection of what a PDE even was so it was like discovering it for the first time. I was intrigued, so I enrolled in a PDE’s math class during the last semester of my undergrad. My professor had a doctorate of art in mathematics, which contrasted greatly with my purely applied engineering background.

At first I scoffed at the professor’s insistence that math was a creative form, but as the class went on I was forced to dig up and actually learn a great deal of the topics that had been presented in earlier math classes. As I did, I began to really understand a few concepts and realize that they actually made sense in and of themselves. Math began to take on a meaning of its own beyond just its applications.

One of the biggest revelations was the ideas of vector spaces and that of variable transformations to map from one vector space to another. After asking my professor for help with just such a problem, I asked how he had known that that particular variable transformation would work. His answer changed the way that I have looked at math ever since. He said “I defined it that way because it was convenient”. The idea was so foreign to me that I had to think about it for some time before it really made sense. I had always thought that there was only one correct variable transformation, one correct proof of every mathematical fact, one best way to solve every problem. The idea that I could pick something, define it, and work out the implications on my own was amazing to me.

My interest in PDE’s led me to begin a study of them on my own. This study has led to a multitude of other topics that are interrelated with PDE’s including higher dimensional vector spaces and other wondrous ideas that lead the imagination to ask a lot of “what if” type questions. One of my most recent reads was a book on the history of mathematics. It was such an eye opener. Math and science progressed hand in hand in most eras (to me, those seemed to be the most fruitful). Many mathematicians were also classified as scientists or experts in other disciplines. Their processes for discovery varied, some liked rigorous proofs, others relied on inspiration/intuition and worked out the details later, some even published erroneous solutions to problems, notations changed and evolved, and creativity flourished. No longer was math a cold, exact, and deterministic subject. It had come to life for me.

I apologize for taking so long to get to my point, but I felt the background was necessary to justify my position since I do not know any educational theories. My point is that I think the best way to balance the applications and the art (the art part is a newly discovered part to me, but I am so glad that I have come to be able to view math in such a way) is to follow the historical development itself. Don’t ask students to solve totally stupid and uninteresting problems about Sally and Rob’s ages, travel distances, etc. Why not use the real questions of the giants that paved the way for us? There are a great deal of artistic and applied problems that spurred the development (and may I add understanding) of mankind. Why rob students of the richness of that history? I think their minds will tend to evolve in understanding over time much the way the ancients did. They can then see our wealth of knowledge as accessible to them. They can ponder on things and question about how the ancients determined the mass of the earth, the percent of gold in a crown, the value of pi or even the fact the ratio of the circumference of a circle to its diameter is constant.

In short, I think the best way to discover math is to relive the discovery of it with a little help from a skilled mentor so as to avoid the intellectual pitfalls of the ancients and of course so it will not take thousands of years to get an education. My opinion is that this would turn math from a dry subject into an interesting narration. It would also give a balance of art and application since neither deserves full credit for the present state of our knowledge.

What if a student will never use it again? My answer to that is that from an art and creativity perspective it will open their mind to consider all the possibilities to any given scenario they may meet and from an applied perspective that it will enhance their appreciation for the world around them just like a study of art an poetry turns a mundane artwork into something with meaning.